tarantello g. selfdual gauge field vortices

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ProgressinNonlinearDifferentialEquationsandTheirApplicationsVolume72EditorHaimBrezisUniversitePierreetMarieCurie´ParisandRutgersUniversityNewBrunswick,N.J.EditorialBoardAntonioAmbrosetti,ScuolaInternationaleSuperiorediStudiAvanzati,TriesteA.Bahri,RutgersUniversity,NewBrunswickFelixBrowder,RutgersUniversity,NewBrunswickLuisCaffarelli,TheUniversityofTexas,AustinLawrenceC.Evans,UniversityofCalifornia,BerkeleyMarianoGiaquinta,UniversityofPisaDavidKinderlehrer,Carnegie-MellonUniversity,PittsburghSergiuKlainerman,PrincetonUniversityRobertKohn,NewYorkUniversityP.L.Lions,UniversityofParisIXJeanMawhin,UniversiteCatholiquedeLouvain´LouisNirenberg,NewYorkUniversityLambertusPeletier,UniversityofLeidenPaulRabinowitz,UniversityofWisconsin,MadisonJohnToland,UniversityofBath GabriellaTarantelloSelfdualGaugeFieldVorticesAnAnalyticalApproachBirkhauser¨Boston•Basel•Berlin GabriellaTarantelloUniversitadiRoma“TorVergata”`DipartimentodiMatematicaViadellaRicercaScientifica00133RomeItalytarantel@mat.uniroma2.itISBN:978-0-8176-4310-2e-ISBN:978-0-8176-4608-0DOI:10.1007/978-0-8176-4608-0LibraryofCongressControlNumber:2007941559MathematicsSubjectClassification(2000):35J20,35J50,35J60,35Q51,58J05,58J38
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PrefaceGaugeFieldtheories(cf.[Ry],[Po],[Q],[Ru],[Fr],[ChNe],[Pol],and[AH])havehadagreatimpactinmoderntheoreticalphysics,astheykeepinternalsymmetriesandcanaccountforimportantphysicalphenomenasuchas:spontaneoussymmetrybreak-ing(seee.g.,[En1],[En2],and[Br]),thequantumHalleffect(seee.g.,[GP],[Gi],[McD],[Fro],and[Sto]),chargefractionalization,superconductivity,andsupergravity(seee.g.,[Wi],[Le],[GL],[Park],[Sch],[Ti],and[KeS]).Inthesenotes,wefocusonspecificexamplesofgaugefieldtheorieswhichadmitaselfdualstructurewhenthephysicalparameterssatisfya“critical”couplingconditionthattypicallyidenti-fiesatransitionbetweendifferentregimes.Theselfdualregimeischaracterizedbythepresenceof“special”soliton-typesolutionscorrespondingtominimizersoftheenergywithincertainconstraintsof“topological”nature.Suchsolutions,knownasselfdualsolutions,satisfyasetoffirst-order(selfdual)equationsthatfurnisha“factorization”forthesecond-ordergaugefieldequations.Furthermore,eachclassof“topologicallyequivalent”selfdualsolutionsformthespaceofmoduli,whosecharacterizationisoneofthemainobjectivesingaugetheory.Thesituationisnicelyillustratedbythe(classical)Yang–Millsgaugefieldtheory(cf.[YM]),whosefieldequations(overS4)aresatisfiedbyselfdual/antiselfdualconnections,simplybyvirtueofBianchiidentity.Theselfdual/antiselfdualconnectionsdefinethewell-knowninstantonsolutionsoftheYang–Millsfieldequations.Theycanbecharacterizedbythepropertythat,amongallconnectionswithprescribedsecondChernnumberN∈Z,theN-instantonsiden-tifythosewithminimumYang–Millsenergy.InthiswayoneseesthateverysecondChern–PontryaginclassofS4canberepresentedbyafamilyofinstantons,whichformsthespaceofmoduliinthiscase.ThespaceofmoduliofN-instantonshasbeencompletelycharacterizedintermsoftheassignedsecondChernnumber,N.Inthisrespect,wereferto[AHS1],[Schw],and[JR]foradimensionalanalysisofsuchspace,andto[BPST],[ADHM],[AHS2],[JNR],[JR],[’tH1],and[Wit1]forexplicitconstructionsofN-instantons.AlthoughinstantonsdonotexhaustthewholefamilyoffiniteactionsolutionsoftheYang–Millsequations(seee.g.,[SSU],[Par],[SS],[Bor],[Bu1][Bu2],and[Ta3]),theirimpactbothinmathematicsandinphysicshasbeenremarkable.Fromthemathematicalpointofview,itisenoughtomentiontheirstrikingimplicationstowardthestudyofdifferentialtopologyforfour-dimensional viPrefacemanifolds(cf.[DK]and[FU]).Intheselfdualcontext,weseethroughtheworkcon-tainedin[Fa],[FM2],[JT],[Le],[Ra],[RS],[AH],[Pe],[Wei],[NO],[Bra1],[Bra2],[GO],[PS],[Hi],and[Y1],howinfluentialhasbeenthestudyofYang–Millsinstan-tonstowardtheunderstandingofotherselfdualsoliton-typeconfigurationsincluding:monopoles,vortices,kinks,strings,etc.Inthisrespect,weshallbeconcernedwiththecasewhere,inaccountoftheHiggsmechanism,weincludeaHiggsfieldinthetheory,tobe(weakly)coupledwiththeothergaugefields.InthiswayweareleadtoconsidertheYang–Mills–Higgstheory,wherewestillmayattainselfdualitybyadimensionalreductionprocedureyieldingtomonopolesasthethree-dimensionalsoliton-solutionsforthecorrespondingselfdualYang–Mills–Higgsequations.Wereferto[JT],[Le],[AH],[Pe],[GO],and[Y1]foradetaileddiscussionofmonopolesbasedontheirstrongtieswithinstantons.AnotherinstructiveexampleaboutselfdualityisofferedbytheabelianHiggsmodelaspointedoutbyBogomolnyiin[Bo].Moreprecisely,byconsideringaplanar(abelian)Maxwell–Higgstheorywithascalar“double-well”potential(hav-ingappropriatestrength),Bogomonlyiin[Bo]derivesasetofselfdualequations,whosesolutionsdescribethewell-knownselfdualMaxwell–Higgsvorticesdiscussedin[NO].Onemayregardtheseconfigurationsasthecross-sectionofthe“vortex-tubes”observedexperimentallyinsuperconductorswhicharesubjectedtoanexternalmag-neticfield.Infact,theselfdualsituationidentifiedbyBogomonlyidescribestherel-ativisticanalogoftheGinzburg–Landaumodelinsuperconductivity(cf.[GL]),withparametersthatcorrespondtotheborderlinecasebetweenTypeIandTypeIIsuper-conductors.InanalogywithYang–Millsinstantons,itispossibletodistinguishabelianMaxwell–Higgsselfdualvorticesintodistinct“topological”classes,relativetoeachel-ementofthehomotopygroupofS1.TheroleofS1inthiscontextisreadilyexplained,sincetopologically,itrepresentstheabeliangaugegroupU(1).Moreprecisely,foreveryN∈π1(S1)=Z,thefamilyofselfdualMaxwell–HiggsN-vorticescorre-spondstotheminimaoftheMaxwell–Higgsenergy,asconstrainedtoHiggsfieldswithtopologicaldegreeN.Moreover,thespaceofmoduliformedbyN-vorticesoverasurfaceMhasbeencompletelycharacterizedasamanifoldequivalenttoMNmod-ulothegroupofpermutationsofNelements(see[Ta1],[JT],[Bra1][Bra2],[Ga1],[Ga2],[Ga3],[WY]and[NO],[MNR],[HJS],[Y1]forfurtherresults).Wementionthat,incontrasttoYang–Millsinstantons,selfdualMaxwell–Higgsvorticesfullyde-scribefiniteenergystaticsolutionsoftheMaxwell–Higgsfieldequations.Inthisway,onededucesacompletecharacterizationofGinzburg–Landauvorticesintheselfd-ualregime.Morerecently,muchprogresshasalsobeenmadeintheunderstandingofGinzburg–Landauvorticesawayfromtheselfdualregime,asonemayseeforexamplein[BBH],[DGP],[Riv],and[PR].Intheearly1990s,Chern–Simonstheorieswereintroducedtoaccountfornewphenomenaincondensedmatterphysics,anyonphysics,supercondictivity,andsupergravity(see[D1],[D3]andreferencestherein).Selfdualityenteredthisnewscenariowithaprimaryrole.Infact,itbecameimmediatelyclearthatalthoughtheChern–Simonstermwasextremelyadvantageousfromthepointofviewofthephysicalapplications,itintroducedseriousanalyticaldifficultiesthatpreventedthecorrespondingChern–Simonsgaugefieldequationsfrombeinghandledwith Prefaceviimathematicalrigor.Thus,togainamathematicalgraspofChern–Simons-Higgscon-figurations,ithasbeenusefultoconsiderconvenientselfdualfirst-orderreductionsofthe(difficult)second-orderChern–Simonsfieldequations.Inthisrespect,followingBogomolnyi’sapproach,Jackiw–Weinberg[JW]andHong–Kim–Pac[HKP]intro-ducedanabelianChern–Simons-Higgs6th-ordermodelthatobeysaselfdualregime.Forsuchamodel,theMaxwellelectrodynamicsisreplacedbytheChern–SimonselectrodynamicsandtheHiggs“double-well”potentialisreplacedbya“triple-well”potential.WeshallseehowthosecharacteristicsgiverisetoaChern–Simonsthe-orywhichsupportsarichclassofselfdualvorticesyettobecompletelyclassified.Subsequently,otherinterestingselfdualChern–Simonsmodelswereintroducedbothinrelativisticandnon-relativisticcontextsandaddresstheabelianandnon-abeliansituation.Werefertotheexcellentpresentationonthissubject,asprovidedbyDunnein[D1]and[D3].ThepurposeofthesenotesistoillustratethenewanddelicateanalyticalproblemsposedbythestudyofselfdualChern–Simonsvortices.Weshallpresenttheresolutionofsomevortexproblemsanddiscussthemanyremainingopenquestions.Bythisanalysis,weshallalsobecomecapableofhandlingthecelebratedelectroweaktheoryinrelationtotheselfdualregimecharacterizedbyAmbjorn–Olesenin[AO1],[AO2],and[AO3].InChapter1,weintroducethebasicmathematicallanguageofgaugetheoryinordertoformulateexamplesofChern–Simons-Higgstheoriesbothintheabelianandthenon-abeliansetting.Wearegoingtocomparetheirfeatureswiththewell-knownabelianHiggsandYang–Mills–Higgsmodel.Forthosetheories,wewillseehowtoattainselfdualityandtoderivetherelativeselfdualequationsthatwillrepresentthemainobjectiveofourstudy.Inthisperspective,weshallinvestigatealsotheelectroweaktheoryofGlashow–Salam–Weinberg(see[La])accordingtotheself-dualansatzintroducedbyAmbjorn–Olesen[AO1],[AO2],and[AO3].Inaddition,weshallanalyzeselfgravitatingelectroweakstrings,astheyoccurwhenwetakeintoaccounttheeffectofgravitythroughthecouplingoftheelectroweakfieldequationswithEinsteinequations.InChapter2,weshalladopttheapproachintroducedbyTaubesfortheabelianHiggsmodel(see[Ta1]and[JT])toreducetheselfdualfieldequationsintoellipticproblemsinvolvingexponentialnonlinearities.Naturally,thiswillleadustoexam-ineLiouville-typeequations,whosesolutions(see[Lio])haveenteredalreadyintheexplicitconstructionofsomespecialselfdualconfigurations,forexample:Wittens’in-stantons(cf.[Wit1]),sphericallysymmetricmonopoles(cf.[JT])andOlesen’speriodicone-vortices,(cf.[Ol]).Unfortunately,ourproblemswillnotbeexplicitlysolvable,andweshallneedtointroducesophisticatedanalyticaltoolsinordertoobtainsolutionswhosefeaturescanbedescribedconsistentlywiththephysicalapplications.Tothisend,weshallrecallsomeknownfactsaboutLiuoville-typeequations,inrelationtotheirmeanfieldfor-mulationsandtotheMoser–Trudingerinequality(cf.[Au]).ThismaterialiscollectedinChapter2togetherwithageneraldiscussiononrelatedmathematicalproblemsandapplications. viiiPrefaceOnthebasisofthisinformation,weproceedinChapter3toanalyzeplanarChern–Simonsvorticesfortheabelian6th-ordermodelproposedbyJackiw–Weinberg[JW]andHong–Kim–Pac[HKP].Here,weencounterthefirstnovelfeatureofChern–SimonsvorticesincomparisonwithMaxwell–Higgsvortices.Infact,wearenowdealingwithatheorythatadmitsbothsymmetricandasymmetricvacuastates,andhenceweexpectmultiplevortexconfigurationstooccuraccordingtothenatureofthevacuumbywhichtheyaresupported.Moreprecisely,intheplanarcasethisamountstoclassifyvorticesinrelationtotheirasymptoticbehavioratinfinity.Thus,weshallcall“topological”thosevorticesthatatinfinityareasymptoticallygaugeequivalenttoanasymmetricvacuumstate;while“non-topological”willbecalledthosethatatinfin-ityareasymptoticallygaugeequivalenttothesymmetricvacuum.ThisterminologyisjustifiedbythefactthatonlythetopologicalsolutionscorrespondtominimizersoftheChern–Simons-HiggsenergywithintheclassofHiggsfieldswithassignedtopologicaldegree.ObservethatsincetheMaxwell–Higgsmodeladmitsonlyasymmetricvacuastates,itcanonlysupporttopologicalvortices.ThegoalofChapter3istoshowrigor-ouslythatindeedboth“topological”and“non-topological”vorticesactuallycoexistintheChern–Simons-Higgstheoryproposedin[JW]and[HKP].Inparticular,inSection3.2weshowthatthetopologicalonesareverymuchequivalenttotheMaxwell–Higgsvortices,withwhomtheysharethesameuniquenesspropertyandasymptoticbehavioratinfinity.Itisinterestingtonotehoweverthatthe“topological”Chern–Simonsvor-ticesarebynomeans“approximations”oftheMaxwell–Higgsvortices(inanyreason-ablesense),asonefindsoutfromsomelimitingproperties(seeproperty(3.1.3.)(c)).Insection3.3weshallpresenttheconstructionofChae–Imanuvilov[ChI1]relativetonon-topologicalChern–SimonsvorticeswhichextendsandcompletesthatofSpruck–Yang[SY1]relativetotheradiallysymmetriccase.Wealsobrieflydescribethealter-nativeconstructionofChan–Fu–Lin[CFL]yieldingtonon-topologicalChern–Simonsplanarvorticessatisfyinganice“concentration”propertyaroundthevortexpoints(i.e.,thezeroesoftheHiggsfield),consistentlytowhathasbeenobservedexperimentallyinthephysicalapplications.However,incontrasttothetopologicalcase,acompleteasymptoticdescriptionof“non-topological”vorticesisstillunderinvestigation.Infact,wearestillfarfromafullcharacterizationofselfdualChern–SimonsvorticesinthesamespiritofwhatisavailablefortheMaxwell–Higgsmodel.Forinstance,weobservethatourstudydoesnotclarifywhethertheChern–Simonsfieldequationsadmit(finiteaction)solutionsotherthantheselfdualones;nordoesitevenjustifythatthiscanneverbethecasewhensomesymmetryassumptionholds,asitoccursforinstantonsandmonopoles.Thesituationisevenlessclearforothermodels,forexample,theabelianMaxwell-Chern–Simons-HiggsmodelproposedbyLee–Lee–Min[LLM]asaunifiedtheoryforbothMaxwell–HiggsandChern–Simons-Higgsmodels.Forthismodeltheasymptoticdistinctionbetweentopologicalandnon-topologicalvorticescarriesover,andtheex-istenceofsuchconfigurationshasbeenestablishedrepectivelyin[ChK1]and[ChI3].ButafullclassificationofselfdualMaxwell-Chern–Simons-Higgsvorticesisstillmissing,includingthevalidityofapossibleuniquenesspropertyforthe“topological”ones.See[ChN]forsomecontributioninthisdirection. PrefaceixInthenon-abelianframework,aninterestingselfdualChern–SimonsmodelhasbeenproposedbyDunnein[D2](seealso[D1]).Vorticesinthiscaseareexpectedtohaveanevenricherstructure,sincenowthesysteminvolves“intermediate”vacuastatesthatinterpolatebetweenthesymmetricandtotallyasymmetricones.Sofar,ithasbeenpossibletoestablishonlytheexistenceofaplanarnon-abelianChern–Simonsvortexoftopologicalnature(see[Y6]).Thedifficultyinthestudyofnon-abelianvor-ticesarisesfromthefactthattheyinvolveellipticproblemsinasystemform(see(2.1.21)and(2.1.25))whichintroducesadditionaltechnicaldifficulties,ascomparedtothesingleLiouville-typeequationarisingfromtheChern–Simons6th-ordermodelof[JW]and[HKP].InChapter4,westillconsidervorticesfortheChern–Simons6th-ordermodelin[JW]and[HKP],butnowweanalyzethemunder(gaugeinvariant)periodicboundaryconditions.Thisismotivatedbythefactthatlatticestructuresarelikelytoforminacondensedmattersystem.Also,theyshouldaccountfortheso-called“mixedstates”,thatAbrikosovdescribedtooccurinsuperconductivity.Healsoanticipatedtheirperi-odicstructurelongbeforetheywereobservedexperimentally(cf.[Ab]).AllperiodicselfdualChern–SimonsvorticescorrespondtominimaoftheChern–SimonsenergyintheclassofHiggsfieldswithassignedtopologicaldegree.There-fore,asfortheabelianHiggsmodel,foranygivenintegerN,themodulispaceofperiodicChern–SimonsN-vorticesisformedbyminimizersoftheenergyamongHiggsfieldsoffixedtopologicaldegreeN.ThisspaceofmoduliturnsouttobemuchricherthanthatoftheMaxwell–Higgsmodeldescribedin[WY].Indeed,weseethatintheperiodiccase,avortexmustapproachavacuumstateastheChern–Simonscouplingparametertendstozero.Again,thepresenceofdifferentvacua(asymmet-ricandsymmetric)givesrisetoasymptoticallydifferentvortexbehaviorsandthisleadstomultiplicity.Thus,inanalogytotheplanarcase,weshallcallof“topological-type”thosevorticesasymptoticallygaugeequivalenttoasymmetricvacuastatesandof“non-topological-type”thosevorticesasymptoticallygaugeequivalenttothesym-metricvacuum,whentheChern–Simonscouplingparametertendstozero.Weshallintroduceausefulcriterion(see[DJLPW])todistinguishbetweenthisdifferentclassofvorticesandshow,thatindeed,bothtypescoexistfortheChern–Simons6th-ordermodelin[JW]and[HKP].Theconstructionof“topological-type”vorticesfollowsbyusingaminimizationprincipleinthesamespiritofthatintroducedforplanartopological(Chern–SimonsorMaxwell–Higgs)vortices.Thisapproachallowsustoclarifywhyauniquenessprop-ertyshouldholdfor“topological-type”vortices,asrecentlyestablishedin[T7].Infact,oneseesthatwheneveravortexisasymptoticallygaugerelatedtoanasymmet-ricvacuumstate,thenfromavariationalpointofview,itmustcorrespondtoalocalminimumfortheassociatedactionfunctional(possiblyforsmallvaluesoftheChern–Simonscouplingparameter).Withthisinformationinhand,itisthenpossibletocheckthatuniquenessmusthold(fordetailssee[T7]andalso[Cho],[ChN]).Moreover,forourvariationalproblemitisalsopossibletocarryouta“mountain-pass”construction(cf.[AR]),andthispermitsustodeducetheexistenceofa“non-topological-type”vortex.Unfortunately,suchconstructionguaranteesconvergenceforthe“non-topological-type”vortextowardthesymmetricvacuumstateonlyin xPrefaceLp-norm,andsoitleavesoutimportantinformationaboutthebehaviorofthevortexsolutionnearthe“vortexpoints.”Thus,toimprovesuchconvergencetoholdatleastinuniformnorm,wediscussanalternativeconstruction(introducedin[T1]and[NT3])thatpermitsustohandlesingleordoublevortices.Interestingly,inthedouble-vortexcasesuchconstructionrelatestheChern–Simonsvortexproblemtoameanfieldequa-tionoftheLiouville-type,whichalsoentersintoothercontextssuchasthestudyofextremalsfortheMoser–TrudingerinequalityortheassignedGausscurvatureproblem(seee.g.,[ChY3]andreferencestherein).Asamatteroffact,our“non-topological-type”double-vortexsolutionscorrespondtothe“best”minimizingsequencefortheMoser–Trudingerinequalityontheflattorus.Thus,tocontroltheirbehavior(astheChern–Simonsparametertendstozero)itisnecessarytodevelopadetailedblow-upanalysisconcerningsolutionsofLiouville-typeequationsinthepresenceof“singular”Diracmeasuressupportedatthevortexpoints.ThistaskwillbethecarriedoutinChapter5.Butbeforeorientingourdiscussiontowardsthosetechnicalanalyticalaspects,weshallcompleteChapter4byextendingourapproachtothestudyofperiodicnon-abelianChern–Simonsvortices.Inthisway,welandnaturallyonthefieldofellipticsystemsoftheLiouville-type,forwhichmuchmoreneedstobeunderstoodandclar-ified.Forthisreason,ourcontributionstotheunderstandingofperiodicnon-abelianChern–Simonsvorticesprovideusonlywithpartialanswersandleaveoutmuchroomforimprovements.Mainly,weshallbeconcernedwiththecaseofSU(n+1)-vortices,whoseanalysisinvolvesthestudyofanellipticToda-latticesystemcharacterizedbymanyelementsofanalyticalinterest(see[JoW1],[JoW2]and[JoLW]).InChapter5wediscusstheblow-upbehaviorofsolution-sequencesforLiouville-typeequationsinthepresenceof“singular”sources.TheaimofthischapteristopresentasystematicextensionoftheworkofBrezis–Merle[BM],Li–Shafrir[LS]Brezis–Li–Shafrir[BLS]andLi[L2]concerningthe“regular”case,namely,whensingularsourcesarenottakenintoaccount.Asitturnsout,thistaskbecomesratherdelicatewhenblow-upoccursatthe“singular”set,asituationlikelytooccurforourvortexsolutions.Inthiscase,thecharacterofthe“concentration”phenomenoniscom-plicatedbymoredegenerateaspects.Nonetheless,itisstillpossibletoobtainsharpconcentration/compactnessprinciples,Harnacktypeinequalities,inf+supestimatesand“quantized”properties,whichfurnisharathercompletedescriptionoftheblow-upphenomenon,asoneexpectstooccurforthevortexsolutions.Thematerialpresentedinthissectionisrathertechnical,andonlyconcernsthecaseofasingleequation.However,thepresenceofsingularsourcesgivesusthechancetointroducemanytechnicaltoolswhichwehopemayhelpintheblow-upanalysisforsystems(bothinthe“regular”and“singular”case),aswellas,forrelatedproblems(cf.[Ci],[CP],[KS],[Mu],[CLMP1],[CLMP2],[Ki1],[Ki2],and[Wo]).InChapter6,wetakeadvantageoftheanalysisdevelopedinChapter5inordertocompletethestudyoftheasymptoticbehaviorofChern–Simonsperiodicdouble-vorticesasconstructedinChapter4.Asabyproductofthisanalysis,weshallbeabletoobtainextremalsfortheMoser–Trudingerinequalityontheflattwo-torus. PrefacexiAlso,theinformationinChapter5,incombinationwithsomevariationaltech-niques,willallowustoestablishageneralexistenceresultforanellipticsystemofinterestinthestudyofselfdualelectroweakperiodicvortices.TheobjectiveofChapter7istoestablishtheexistenceofselfdualelectroweakcon-figurationsofAbrikosov’s“mixed-type”vorticesandofself-gravitatingelectroweakstrings.WewillfollowthepathopenedinthestudyofChern–Simonsvorticestoob-tainresultsinelectroweaktheorybymeansofthemethodsandtechniquesintroducedinthepreviouschapters.Thereagain,ouranalysissufficestohandleonlyacertainrangeofparameters,anditwouldbeextremelyusefultoknowwhetherornotourresultsextendtocoverthefullrangeofadmissibleparameters.Inthismonograph,wehavechosentodiscussonlyafewselectedselfdualmodelswhichinourview,mosteffectivelyillustratetheadvantageoftheanalyticalapproachthatispursuedhereandthatoriginatedintheworkofTaubes([Ta1]and[JT]).Asalreadymentioned,thislineofinvestigationhasprovedequallysuccessfulinthestudyofmanyotherselfdualgaugefieldconfigurations.However,forthemodelsconsideredhere,theprogressachievedintheselfdualcaseisparticularlyremarkablesincerigor-ousmathematicalresultsawayfromtheselfdualregimeremainratherscarce(seee.g.,[HaK],[KS1],and[KS2]).Whilewehaveprovidedindicationsonpossibleextensionsofthegivenresultstorelatedmodels,werefertothemonographofYang[Y1]forasystematicuseofTaubes’strategytotreatselfdualsolutionsarisingindifferentphysicalcontexts.Infact,wehopethatthereadercanprofitfromtheanalysisdevelopedhereandfurtherpursuetheinvestigationofavarietyofselfdualmodels,aftertheworkin[Y1].Wehavenottoucheduponotheraspectsrelatedtoselfduality,forexample:“inte-grability”issues(seee.g.,[Das],and[Hop]),dynamicalpropertiesofseldfualvortices(seee.g.,[KL],[Ma],and[BL]),andsolvabilityofinitialvalueproblems(seee.g.,[Ch5],[ChC],and[ChCh2]).Alloftheseproblemsposeveryattractiveandstimulat-ingmathematicalquestions,andcertainlydeservealotmoreattention.Acknowledgments.ItisapleasuretothankHaimBrezisforhisencouragementtotake´onthisprojectandforhisinterestandcontinuedsupport.Also,wehavebenefitedfromtheusefulobservationsoftherefereesandprofitedfromthevaluablecommentsofPierpaoloEsposito. ContentsPreface..........................................................v1SelfdualGaugeFieldTheories..................................11.1Introduction.................................................11.2TheabelianMaxwell–HiggsandChern–Simonstheories...........21.3Non-abeliangaugefieldtheories................................131.3.1Preliminaries..........................................131.3.2Theadjointrepresentationandsomeexamples..............141.3.3Gaugefieldtheories....................................171.3.4TheCartan–Weylgenerators:basics......................191.3.5Yang–Mills–Higgstheory...............................261.3.6Aselfdualnon-abelianChern–Simonsmodel...............301.4Selfdualityintheelectroweaktheory............................352EllipticProblemsintheStudyofSelfdualVortexConfigurations.....432.1Ellipticformulationoftheselfdualvortexproblems................432.2ThesolvabilityofLiouvilleequations............................522.3Variationalframework........................................562.4Moser–Trudingertypeinequalities..............................572.5AfirstencounterwithmeanfieldequationsofLiouville-type........642.6Finalremarksandopenproblems...............................723PlanarSelfdualChern–SimonsVortices..........................753.1Preliminaries................................................753.2PlanartopologicalChern–Simonsvortices........................783.3Auniquenessresult...........................................993.4Planarnon-topologicalChern–Simonsvortices....................1113.5Finalremarksandopenproblems...............................1274PeriodicSelfdualChern–SimonsVortices.........................1314.1Preliminaries................................................1314.2Constructionofperiodic“topological-type”solutions..............134 xivContents4.3Constructionofperiodic“non-topological-type”solutions..........1444.4Analternativeapproach.......................................1504.5MultipleperiodicChern–Simonsvortices........................1624.6Finalremarksandopenproblems...............................1715TheAnalysisofLiouville-TypeEquationsWithSingularSources.....1735.1Introduction.................................................1735.2Backgroundmaterial..........................................1755.3Basicanalyticalfacts..........................................1805.4Aconcentration-compactnessprinciple..........................1845.4.1Theblow-uptechnique.................................1845.4.2Aconcentration-compactnessresultarounda“singular”point......................................1925.4.3Aglobalconcentration-compactnessresult.................1985.5Aquantizationpropertyintheconcentrationphenomenon..........1995.5.1Preliminaries..........................................1995.5.2AversionofHarnack’sinequality........................2005.5.3Inf+Supestimates....................................2065.5.4AQuantizationproperty................................2135.5.5Examples.............................................2205.6Theeffectofboundaryconditions...............................2255.6.1Preliminaries..........................................2255.6.2Pointwiseestimatesoftheblow-upprofile.................2275.6.3Theinf+supestimatesrevised...........................2375.7Theconcentration-compactnesssprinciplecompleted..............2415.8Finalremarksandopenproblems...............................2476MeanFieldEquationsofLiouville-Type..........................2496.1Preliminaries................................................2496.2Anexistenceresult...........................................2536.3ExtremalsfortheMoser–Trudingerinequalityintheperiodicsetting.........................................2616.4TheproofofTheorem4.4.29...................................2746.5Finalremarksandopenproblems...............................2787SelfdualElectroweakVorticesandStrings........................2817.1Introduction.................................................2817.2Planarselfgravitatingelectroweakstrings........................2847.3TheproofofTheorem7.2.2....................................2887.4Periodicelectroweakvortices...................................3007.5Concludingremarks..........................................303References.......................................................305Index............................................................323 1SelfdualGaugeFieldTheories1.1IntroductionInthischapterweintroducethereadertothegaugetheoryformalisminordertofurnishexamplesofgaugefieldtheoriesthatsupportaselfdualstructure.Westartwiththesimplerabeliansituation,wheremostofthetechnicalaspectsofgrouprepresentationtheorycanbeavoided.Fromthephysicalpointofview,anabeliangaugefieldtheorydescribeselectro-magneticparticleinteractions.ThusweshallstartbydiscussingtheabelianMaxwell–Higgsmodel,well-knownalsoastherelativisticcounterpartoftheGinzburg–Landaumodelinsuperconductivity(cf.[GL]).WewillillustrateBogomolnyi’sapproach(cf.[Bo])andattainselfdualityforthismodelwithparametersthatdescribethebor-derlinecasethatdistinguishesbetweentypeIandtypeIIsuperconductors.Nextwewillseehow,inthesamespirit,onecanattainselfdualityinthepres-enceoftheChern–Simonsterm(cf.[D1]).Inthiscontext,wewillfocusonthe“pure”Chern–Simons6th-ordermodelofJackiw–Weinberg[JW]andHong–Kim–Pac[HKP]andontheMaxwell–Chern–Simons–HiggsmodelofLee–Lee–Min[LLM].Subse-quently,wewillturntothetreatmentofnon-abeliangaugetheories,andforthispur-poseweshallneedtorecallsomebasicfactsabouttherepresentationofcompact(semisimple)Liegroups(see[Ca],[Hu],and[Fe]).Inthenon-abelianframework,weshallformulatetheYang–MillsandtheYang–Mills–Higgstheories,aswellasthenon-abelianChern–Simonstheoryin[D2]andthecelebratedelectroweaktheoryofGlashow–Salam–Weinberg(cf.[La]).Wewillshowhowtoattainselfdualityforsuchnon-abelianmodels,whichrepresentonlya“sample”oftheamplelistofselfdual(relativistic)gaugefieldtheoriesavailableinphysicsliterature.Someextensionsofthemodelsconsideredherearecontainedin[KLL],[KiKi],[Kh],[CaL],[CG],[Wit2],[Va],and[D1].Inanycase,werefertheinterestedreadertothemonographbyYang([Y1])forabroaderdiscussionofrelativisticandnon-relativisticselfdualtheoriesthatmodelawiderangeofphysicalphenomena. 21SelfdualGaugeFieldTheories1.2TheabelianMaxwell–HiggsandChern–SimonstheoriesTheabelianMaxwell–Higgs(orsimplyabelian-Higgs)andChern–Simonstheoriesdescribeelectromagneticinteractionsand,asgaugefieldtheories,theyareformulatedbyaLagrangeandensityLexpressedintermsofthegaugepotentialAandtheHiggs(matter)fieldφ.Occasionallyaneutralfieldisalsoincluded.WefocusourattentiononeuclideantheoriesformulatedovertheMinkowskispace(R1+d,g)withmetrictensorg=diag(1,−1,...,−1),wherewedenotebyx0thetimevariableand(x1,...,xd)thespacevariables.Usually,weshalluseGreekindicestoidentifyindifferentlyspaceortimevariables,whiletheromanletterswillbespecifictospacevariables.Weadoptstandardnotationsandusethematricesg=(gαβ)intheusualwaytoraiseorlowerindices,andletg−1=(gαβ)fortheinverse.Moreoverweusethesummationconventionoverrepeatedlowerandupperindices.Inthiscontext,thepotentialfieldAisspecifiedbyitssmoothrealcomponentsA1+d→R,α=0,1,...,d;(1.2.1)α:Rwhereas,theHiggsfieldφisasmoothcomplexvaluedfunctionφ:R1+d→C.(1.2.2)Tobeconsistentwiththetheoreticalgauge-formalism(seenextsection),weidentifythepotentialfieldAwitha1-form(connection)asA=−iAα,(1.2.3)αdxandweexpressthecorresponding(Maxwell)gaugefieldFAasthe2-form(curvature)iαβFA=−Fαβdx∧dx,(1.2.4)2whereFαβ=∂αAβ−∂βAα.(1.2.5)TheHiggsfieldφin(1.2.2)isweaklycoupledwiththepotentialfieldAviatheexteriorcovariantderivativeDAasfollows:αwhereDDAφ=Dαφdxαφ=∂αφ−iAαφandα=0,1,...,d.(1.2.6)WerecordtheBianchiidentity∂γFµν+∂µFνγ+∂νFγµ=0,(1.2.7)whichisvalidforthe(curvature)componentsFαβin(1.2.5).Forlateruse,wepointoutthatinthedimensiond=3,identity(1.2.7)maybemoreconvenientlyexpressedintermsofthedualgaugefield:F˜iF˜αβ1µνA=−αβdx∧dxwhereF˜αβ=εαβµνF22 1.2TheabelianMaxwell–HiggsandChern–Simonstheories3asfollows:∂αββF˜=0,α=0,1,2,3.(1.2.8)RecallthatεαβγνdenotestheusualLevi-Civitaε-symbolwhichistotallyskew-symmetricwithrespecttothepermutationofindicesandisfixedbythecondition:ε0123=1.Innormalizedunits,theabelianMaxwell–HiggsLagrangeandensitytakestheform
1αβ1α2L(A,φ)=−FαβF+DαφDφ−V|φ|,(1.2.9)42wherethescalarpotentialVistakenasthefamiliar“double-well”potentialλ
2V(|φ|2)=|φ|2−1,(1.2.10)8withλ>0aphysicalparameter.TheinternalsymmetriestypicalofelectromagneticinteractionsareexpressedbythefactthatthefieldsAandφaredefinedonlyuptothefollowinggaugetransformationsφ→eiωφ,(1.2.11)A→A−idω,(1.2.12)foranysmoothrealfunctionωoverR1+d.TheinvarianceofLunderthetranformations(1.2.11)and(1.2.12)canbeeasilyverified.Infact,thevalidityofsuchinvariancepropertyservesasjustificationforthestructureofLin(1.2.9).Clearly,thesameinvarianceunder(1.2.11)and(1.2.12)ismantainedbythecor-respondingEuler–Lagrangeequationsµ∂VDµDφ=−2,(1.2.13)∂φ∂µν=Jµ,(1.2.14)νFwhere
µiµµJ=φDφ−Dφφ.(1.2.15)2NotethatJµcanbeconsideredasthecurrentgeneratedbytheinternalsymmetriesexpressedby(1.2.11)and(1.2.12).Infact,inviewof(1.2.13),Jµdefinesaconservedquantity,thatis,∂µJµ=0.Furthermore,byidentifyingρ=J0andj=Jµ(1.2.16)withthechargedensityandthecurrentdensityrespectively,weseethat(1.2.8)and(1.2.14)formulatethefamiliarMaxwell’sequationsintermsoftheelectricfieldE=(Ej)andthemagneticfieldB=(Bj)specifiedasfollows:0j1jklEj=−F,Bj=−εFkl(1.2.17)2(εjkl=ε0jkl;j,k,l=1,2,3). 41SelfdualGaugeFieldTheoriesInparticular,intheabsenceofthematterfield(i.e.,φ=0),Jα=0and(1.2.14)reducestoMaxwell’sequationsinavacuum:∂µν=0.(1.2.18)νFFordetailssee[JT]and[Y1].ThefieldsAandφarenotobservablequantities,astheyaredefinedonlyuptothegaugetransformations(1.2.11)and(1.2.12).Onthecontrary,theelectricandmagneticfields(1.2.17)aswellasthemagnitude|φ|oftheHiggs(matter)fieldaregauge-independentquantities,andhenceobservables.Therefore,fromananalyticalpointofview,wecanhopetoexplicitlysolve(1.2.13)and(1.2.14)onlyintermsofthosegauge-invariantquantities.Weshallbeinterestedinobtaining“soliton”configurations,namely,staticsolu-tionsfor(1.2.13)and(1.2.14)carryingfiniteenergy.Tothisend,notethatbytheGausslawconstraint(i.e.,theµ=0componentof(1.2.14))
0ji00∂jF=φDφ−φDφ,(1.2.19)2weeasilyobtainthattheenergydensityassociatedtoL∂L∂L∂LE=∂0Aµ+∂0φ+∂0φ−L(1.2.20)∂∂0Aµ∂(∂0φ)∂∂0φ(inthetemporalgauge)takestheform:1212121D2E=F0j+Fjk+|D0φ|+jφ+V.(1.2.21)2422Ourinterestinstaticconfigurationsismotivatedbythefactthatinanon-relativisticcontext,whenthedimensionisd=3,theLagrangeandensity(1.2.9)withthescalarpotentialVof(1.2.10)hasbeenproposedasamodelforsuperconductivityaccord-ingtotheGinzburg–Landautheory(cf.[GL]).Inthiscontext,φplaystheroleoftheorderparameter,whosemagnitude|φ|measuresthenumberdensityofCooperpairs.Thus,thesuperconductivestateismanifestedwhere|φ|takesvaluesawayfromzero.Furthermore,theconstantλinvolvedinthe“double-well”scalarpotential(1.2.10)definesarelevantphysicalparameterinthiscontext,asitdistinguishesbetweensuper-conductorsofTypeI(i.e.,λ<1)andofTypeII(i.e.,λ>1).Byconsideringacrosssectionofthesuperconductivebulkofamaterial,a“special”situationoccurswhenweconsiderbi-dimensionalsolitonsolutions(vortices)of(1.2.13)and(1.2.14)atthe“critical”couplingvalueofλ=1.Indeed,from(1.2.21)weseethatford=2andλ=1,thestaticenergydensityofthevortex-configurationsinthetemporalgauge(whichallowsustotakeA0=0)coincideswiththeoppositeoftheactionfunctionalLstaticsothatittakestheform,1111
2E2222static=−Lstatic=F12+|D1φ|+|D2φ|+|φ|−1.(1.2.22)2228 1.2TheabelianMaxwell–HiggsandChern–Simonstheories5Thus,ifweintroducetheoperatorsD±φ=D1φ±iD2φ(tobecomparedwiththe∂¯and∂operatorsofcomplexfunctionsinagauge-freeset-ting)andobservethat2+|D222jk|D1φ|2φ|=|D±φ|±F12|φ|∓ε∂jJk,(1.2.23)wearriveatthefollowingusefulexpressionforthestaticenergy:
21211211jkEstatic=|D±φ|+F12±|φ|−1±F12∓ε∂jJk.(1.2.24)22222Againεjkisthe(bi-dimensional)skew-symmetricε-symbol(j,k=1,2),whichwecanobtainfromεαβγνdefinedabovesimplybysettingεjk=ε0jk3.So,byconsideringboundaryconditionssuitabletoneglectingthetotalspatialdivergencetermsin(1.2.24),wefindthatenergyminimizervorticesmustsatisfythefollowingfirst-orderequations:D±φ=0,(1.2.25)
2F|φ|2−1=0,(1.2.26)12±A0=0.(1.2.27)Solutionsof(1.2.25)–(1.2.27)areknownastheNielsen–Olesenvortices(cf.[NO])andareenergyminimizersconstrainedtotheclassofgaugepotentialfieldswithafixedmagneticflux.Weshallseethatthiscorrespondstoa“topological”constraint,suchtoproduce“quantization”effects.Equations(1.2.25)–(1.2.27)werederivedbyBogomolnyiin[Bo]asaconvenientfirst-orderfactorizationofthesecond-orderEuler–Lagrangeequations(1.2.13)and(1.2.14),wherethescalarpotentialVsatisfies(1.2.10)withλ=1.Indeed,itisasimpletasktocheckthateverysolutionof(1.2.25)–(1.2.27)alsosatisfies(1.2.13)and(1.2.14).Suchareductionpropertyhasbeenobservedtooccurinquiteavarietyofmod-elsingaugefieldtheory,whenthephysicalparametersarespecifiedaccordingtoanappropriate“critical”coupling.Thefirstinstanceofsuchanoccurancehasbeenobservedinthenon-abeliancontextforthepureYang–Millsmodel(cf.[JT]).Inthiscase,energyminimizers(withinatopologicalclass)giverisetoinstantonsandcorre-spondtoselfdual(orantiselfdual)connections,asdiscussedinSection1.3.5.Byanalogy,ithasbecomeacustomtorefertothereducedfirst-orderequationsastheselfdual(antiselfdual)equations.Wecanrevealtheselfdual/antiselfdualcharacterof(1.2.25)ifweexpressitintheform:Djφ=∓iεjkDkφ.(1.2.28)However,beforeengagingwithnon-abelian(selfdual)gaugefieldtheories,letusseehowasimilarreductionprocedurecanalsobeattainedwhenweenrichtheelectro-dynamicalpropertiesofthetheorybyincludingtheChern–Simonsterm. 61SelfdualGaugeFieldTheoriesTheChern–Simonstheoryisaplanartheory(i.e.,d=2,ormoregeneraldiseven)thatenjoysseveralfavorablephysicalpropertiesnotattainablethroughthe“conventional”Maxwellelectrodynamics.Forinstance,weshallseethatMaxwell–Chern–Simonsvortex-configurationscarrybothelectricalandmagneticcharge,incon-trastwiththeconventionalMaxwellelectrodynamics,thatonlyyieldstoelectricallyneutralGinzburg–Landauvortices.Thisandotherimportantaspectsof(relativisticandnon-relativistic)Chern–SimonstheoriesarewidelydiscussedbyDunnein[D1],inrelationstotheirrelevanceinhighcriticaltemperaturesuperconductivity,thequan-tumHalleffect,conformalfieldtheoryandplanarcondensedmatterphysics.Intheabeliancontext,theChern–SimonsLagrangeandensityLcsisassignedinR1+2intermsofthepotentialfieldA=−iAαdxα,α=0,1,2as1αβγLcs(A)=εAαFβγ,(1.2.29)4whereagainεαβγdenotesthetotallyskew-symmetricpseudotensorfixedbysettingε012=1.ThestructureoftheChern–SimonsLagrangeanLcsmaybejustifiedonthebasisofareductionargumentfromthe4-dimensionalYang–Millsequationstodimensiond=2(see[D1]fordetails).Inthisrespect,observethatLcscorrespondstotheactionfunctionalforthe(trivial)equationFαβ=0(1.2.30)sincewehave∂Lcs1µαβ=εFαβ.(1.2.31)∂Aµ4WeremarktheinterestingfactthatalthoughLcsisnotgauge-invariant,1Lµαβcs(A−idω)=Lcs(A)+∂µωε∂αAβ;2thecorrespondingEuler–Lagrangeequation(1.2.30)isgauge-invariant,andforthisreason,LcsisanadmissibleLagrangeaninthecontextofgaugefieldtheory.Asafirstexample,wedescribeatheoryproposedbyJackiw–Weinberg[JW]andHong–Kim–Pac[HKP]inwhichtheelectrodynamicsofthesystemisgovernedsolelybytheChern–SimonsLagrangean.ThecorrespondingChern–Simons–HiggsLagrangeandensitytakestheform:
kαβµα2L(A,φ)=−εAαFβµ+DαφDφ−V|φ|,(1.2.32)4wherethe(dimensionless)couplingconstantk>0measuresthestrengthoftheChern–Simonsterm,whichweshallrefertoastheChern–Simonsparameter. 1.2TheabelianMaxwell–HiggsandChern–Simonstheories7TheEuler–Lagrangeequationsrelativeto(1.2.32)areexpressedasDαφ=−∂V,(1.2.33)αD∂φkεµαβFµαβ=J,(1.2.34)2whereJµ=iφDµφ−φDµφ.(1.2.35)Again,Jµcanbeconsideredastheconservedcurrentforthesystem,withρ=J0thechargedensityandj=Jkthecurrentdensity.Asbefore,usingtheGausslawconstraintobtainedfromtheµ=0componentof(1.2.34),
kF0012=iφDφ−φDφ,(1.2.36)weeasilydeducethatfortheLagrangeandensity(1.2.32),theassociateenergydensitytakestheexpression
2+|D22E=|D0φ|jφ|+V|φ|(providedweneglectthetotaldivergenceterm).SinceLcsisindependentoftheMinkowskimetric,wecheckthatindeed,itdoesnotcontributetotheenergymomentumtensor.Thus(pure)Chern–Simonsvorticeswillcorrespondtosolutionsfor(1.2.33)and(1.2.34)independentofthex0-variableandwithfinite(static)energy.Notethat,theGausslawconstraint(1.2.36)forthetime-independentcasereducesto2=JkF12=2A0|φ|0.(1.2.37)Identity(1.2.37)canbeusedtogetherwiththeidentityεjk2+|D222|D1φ|2φ|=|D±φ|±F12|φ|∓∂jJk(1.2.38)2(theequivalentof(1.2.23))toobtainthegauge-invariantpartoftheenergydensityrelativetovortexconfigurationsas
jk2222εEstatic=|A0φ|+|D±φ|±F12|φ|+V|φ|∓∂jJk
2(1.2.39)2εjk=|D24222±φ|±A0|φ|+A0|φ|+V|φ|∓∂jJk.k2AsfortheabelianMaxwell–Higgsmodel,wecanoperateonasuitablechoiceofthescalarpotentialVinordertocompletethesquarein(1.2.39)andidentifyasetoffirst-order(selfdual)equationsthat“factorizes”(1.2.33)and(1.2.34).Toaccomplishthisgoalandaccountforsomephysicalconsistency,Jackiw–Weinbergin[JW]andHong–Kim–Pacin[HKP]proposedthetriple-wellpotential
1
2V|φ|2=|φ|2|φ|2−ν2,(1.2.40)k2 81SelfdualGaugeFieldTheorieswiththemass-scalesymmetry-breakingparameterν2.FortheabelianMaxwell–Higgsmodel(1.2.9)wehaveadoptedthenormalizationν2=1.Alsonoticetheexplicitdependenceofthe(selfdual)scalarpotential(1.2.40)ontheChern–Simonsparameterk.ThereforeasfortheabelianMaxwell–Higgsmodel,weseethataselfdualregimeisreachedwhenthereisabalancebetweenthestrengthoftheelectromagneticterm(MaxwellorChern–Simons)andthepotentialterm.FortheChern–Simons6th-ordermodelproposedin[JW]and[HKP],k1
2αβγAα222L(A,φ)=−εαFβγ+DαφDφ−|φ||φ|−ν,(1.2.41)4k2the(static)energydensityofvortex-configurationstakestheform
2jk221222εEstatic=|D±φ|+|φ|A0±|φ|−ν±νF12∓∂jJk.(1.2.42)k2Consequently,byconsideringsuitableboundaryconditionsthatallowonetoneglectthetotalspatialdivergencetermin(1.2.42),wearriveatthefollowingfirst-orderequa-tionstobesatisfiedbyenergyminimizers(atfixedflux)Chern–Simonsvortexsolu-tions,togetherwiththeGauss-lawconstraint(1.2.37)D±φ=0,(1.2.43)
122|φ|(A0±|φ|−ν=0.(1.2.44)kWecanarrange(1.2.37),(1.2.43),and(1.2.44)moreconvenientlyinthefollowingequivalentsetofselfdualequations:⎧⎪⎨D±φ=0,F12=±2|φ|2ν2−|φ|2,(1.2.45)⎪⎩k22A0|φ|2=kF12.Again(1.2.45)representsa“factorization”ofthesecond-orderEuler–Lagrangeequations(1.2.33)and(1.2.34).Indeed,onecaneasilycheckthatasolutionof(1.2.45)alsoverifies(1.2.33)and(1.2.34)withVspecifiedby(1.2.40).Inparticularobservethatawayfromthezerosofφ,theA0-componentofthepotentialfieldisdeterminedsimplybytheidentity:
122A0=±ν−|φ|(1.2.46)k(see(1.2.44)).Thus,theselfdualvortexsolutioniscompletelyidentifiedbythecom-ponents(A1,A2,φ),satisfyingthefirsttwoequationsin(1.2.45).ItisinterestingtonotethattheChern–Simonsenergydensitytakesasimilarstruc-ture(i.e.,sumofthequadratictermsplusthespatialdivergenceterms)forthetime-dependentsolutionsof(1.2.33)and(1.2.34),providedwespecifythescalarpotentialVasin(1.2.40). 1.2TheabelianMaxwell–HiggsandChern–Simonstheories9Infact,the(non-static)Gausslawconstraint(1.2.36)maybeusedtogetherwith(1.2.38)todeducethefollowingexpressionfortheenergydensity:
22jkEcs=D0φ∓iφ|φ|2−ν2+|D±φ|2±νJ0∓ε∂jJk.(1.2.47)kk2Asnotedabove,forfixedmagneticflux(see(1.2.37)),energyminimizersmaybeidentifiedassolutionsofthefirst-orderequationsD±φ=0,(1.2.48)
0i22Dφ=±φ|φ|−ν,(1.2.49)ktobesatisfiedinadditiontotheGausslawconstraint(1.2.36).Equivalently,byinsert-ingequation(1.2.49)into(1.2.36)forthetime-dependentcase,weobtainthefollowingChern–Simonsselfdualequations:⎧⎪⎨D±φ=0,F12=±2|φ|2ν2−|φ|2,(1.2.50)⎪⎩k2D0φ=±iφ|φ|2−ν2.kAsoliton-likesolutionof(1.2.50)maybeconstructedoutofasolution(A0,A1,A2,φ)static,ofthestaticselfdualChern–Simonsequations(1.2.45),simplybylettingiω2x0±φ=ekφstatic,1
(1.2.51)2+ω2−|φ2Aj=(Aj)static,A0=±νstatic|,kforanyconstantω∈R.Inviewof(1.2.46)thesoliton-likesolutionabovereducestotheselfdualvortex(A0,A1,A2,φ)static,whenω=0.Nextweseehowananalogousselfdualreductionpropertyremainsvalidbyconsid-eringafullMaxwell–Chern–Simons–Higgstheory(MCSH-theory),thatalsoincludesaneutralscalarfield.WeshalldiscussamodelproposedbyLee–Lee–Min[LLM]withthepurposeofunifyingtheabelianMaxwell–HiggsandChern–Simons-Higgstheoriesconsideredabove.Tothispurpose,itisconvenienttointroducetheexplicitdependenceofthetheoryintermsoftheelectricchargeq(previouslynormalizedto1)sothatthe(dimension-less)Chern–Simonsparameterisexpressedasσk=,(1.2.52)q2withσhavingthedimensionofmass. 101SelfdualGaugeFieldTheoriesTheMCSH-theoryproposedin[LLM]isformulatedbymeansoftheLagrangeandensity1αβσαβγαL(A,φ,N)=−FαβF−εAαFβγ+DαφDφ4q24q2(1.2.53)
1α2+∂αN∂N−V|φ|,N,2q2whereNisaneutralscalarfield,andthescalarpotentialVtakestheform
22q2ν2q2σV|φ|2,N=|φ|2N−+|φ|2−N,(1.2.54)σ2q2withthemassscaleparameterν2tobeconsideredasasymmetry-breakingparameter.Notethatformally,wemayrecovertheabelianMaxwell–HiggsLagrangeanden-sity(1.2.9)outof(1.2.53)bylettingσ→0whilekeepingqfixed.InthiswaytheChern–SimonstermdropsoutwhiletheneutralscalarfieldNneedstobefixedac-cordingtotherelationσN2=ν.q2Then(1.2.53)reducesto1q2
2LAH(A,φ)=−Fαβα22αβF+DαφDφ−|φ|−ν,(1.2.55)4q22whichgivesexactlytheselfdualabelianMaxwell–HiggsLagrangeandensitywithallrelevantphysicalparameters.Ontheotherhand,ifweletσ→+∞,q→+∞whilekeepingfixedtheChern–Simonsconstantgivenbytheratio(1.2.52),weagainformallyseethatboththeMaxwelltermandthekinetictermrelativetotheneutralfielddropout,whileNmustbefixedaccordingtotherelationσN=|φ|2.Thusatthelimit,Lin(1.2.53)q2takesexactlytheformoftheChern–SimonsLagrangean(1.2.32).ItisinthissensethatwesaythattheMCSH-Lagrangeanin(1.2.53)formulatesaunifiedtheorybetweentheabelianMaxwell–HiggsandChern–Simonsmodels.The“formal”limitstakenabovecanbeshowntoholdrigorouslyalongvortexconfigurations(see[RT1]).LetusnowdescribetheselfdualstructureoftheMCSH-Lagrangeanrepresentedby(1.2.53)and(1.2.54).Firstofall,noticethatthecorrespondingEuler–Lagrangeequationscompletethosein(1.2.34)and(1.2.33)aswehave:⎧⎪⎪1∂α∂αN=−∂V|φ|2,N,⎨q2∂NDαφ=−∂V|φ|2,N,αD(1.2.56)⎪⎪∂φ⎩1∂µνkµαβµq2νF+2εFαβ=J,withkin(1.2.52)andJµthecurrentdefinedin(1.2.35). 1.2TheabelianMaxwell–HiggsandChern–Simonstheories11Asabove,theµ=0componentofthelastequationin(1.2.56)expressestheGausslawconstraintforthegivensystem,whichisgivenasfollows:
1∂0j000jF+kF12=J=iφDφ−φDφ.(1.2.57)q2Asbefore,weshalltakeadvantageof(1.2.57)and(1.2.38)inordertoexpressthefollowingMCSH-energydensity:121222212E=|F0j|+|F12|+|D0φ|+|D1φ|+|D2φ|+|∂0N|2q22q22q22222(1.2.58)122q2q2σ+|∇N|+|φ|N−ν+|φ|−N.2q2σ2q2(neglectingatotalspatialdivergenceterm)inthemoreconvenientform:121E=0j±∂22FjN+|∂0N|+|D±φ|2q22q2221σν2+F22−N+N−(1.2.59)12±q|φ|D0φ∓iφ2q2q2kν2εjk1±0∓∂JjJk+NF0j.k2q2Therefore,withthehelpofsuitableboundaryconditionsthatallowonetoneglectthelasttotalspatialdivergencetermin(1.2.59),theminimalMCSH-energyforfixedflux,issaturatedbythesolutionsofthefollowingselfdualequations:⎧⎪⎪D±φ=0,⎪⎪⎪⎪F0j±∂jN=0,⎨F12±q2|φ|2−kN=0,(1.2.60)⎪⎪
2⎪⎪D0φ∓iφN−ν=0,⎪⎪k⎩∂0N=0,withkgivenin(1.2.52).Oncemore,equations(1.2.60)supplementedwiththeGausslawconstraint(1.2.57)identifyafirst-orderfactorizationofthesecond-orderEuler–Lagrangeequations(1.2.56).Inparticular,selfdualMCSH-vorticeswillbeobtainedbyconsideringsolutionsof(1.2.60)independentofthex0-variable.Inthissituation,thecorrespondingequationssimplifyconsiderably,sincethelastequationisautomaticallysatisfied.Wecansatisfythesecondandfourthequationin(1.2.60)simplybysetting:ν2A0=±−N.(1.2.61)k 121SelfdualGaugeFieldTheoriesFurthermore,inthex0-independentcasetheGausslawconstraint(1.2.57)takestheform12A0+kF12=2A0|φ|.(1.2.62)q2Therefore,wecancombine(1.2.61),(1.2.62),andthethirdequationin(1.2.60)toseethataselfdualMCSH-vortexiscompletelydeterminedintermsofthecomponents(A1,A2,φ,N)satisfying:⎧⎪⎪D±φ=0,⎨F12=±q2kN−|φ|2,(1.2.63)⎪⎪
2⎩−1N=2|φ|2ν−N+kq2|φ|2−kN.q2kExactlyasbefore,weseethatfromasolution(A1,A2,φ,N)staticof(1.2.63),wemayobtainasoliton-likesolutionof(1.2.60)byletting:iω2x0±φ=ekφstatic,Aj=(Aj)static,j=1,2,
122N=Nstatic,andA0=±ν+ω−Nstatic,kforeveryω∈R.Analogousselfdualreductionproceduresareknowntoholdalsofornon-relativisticversionsoftheMaxwellandChern–Simonstheoriesdescribedabove,whereroughlyspeaking,thecovariantderivativeD0φoftheHiggsfieldonlyenterslinearlyintotheLagrangeandensity.Wereferto[D1]foradetaileddiscussioninthisdirection.Alreadyfromthosefirstexamples,wecanremarkonsomeinterestingfeaturescommontoallselfdualequationsdiscussedsofar.Firstly,allofthemincludetheself-dual/antiselfdualequationD±φ=0,(1.2.64)whichwillplayacrucialroleintheanalysisthatfollows.Forthemoment,letusmentionthat(1.2.64)maybeviewedasagauge-invariantversionoftheCauchy–Riemannequation,andinfactitimpliesanholomorphic-typebehaviorfortheHiggsfieldφ(respectivelyφ)uptogaugetranformations(cf.[JT]).Notealsothatexceptfor(1.2.64),theremaining(static)selfdualequationsinvolveonlygauge-invariantquanti-ties(i.e.,F12,|φ|2,andwhenpresent,theneutralscalarfieldN).Therefore,onemayhopetofindanappropriategaugetransformationaccordingtowhichthefullsetofselfdualequationstakethemostconvenientexpressionfromtheanalyticalpointofview.ThisgoalwasattainedfirstbyTaubes(see[Ta1],[Ta2],and[JT])whosuccess-fullyhandledtheselfdualabelianMaxwell–Higgsmodel.Itisoneofourpurposestoshowthat,infact,Taubes’approachworksequallywellfortheChern–Simonsmodelsdiscussedaboveand,moregenerally,forthenon-abeliantheoriesofnextsection. 1.3Non-abeliangaugefieldtheories13Butbeforetreatingnon-abeliangaugefieldtheories,wewishtomentionthatwhileTaubes’approachhasfurnishedacompletecharacterizationofGinzburg–Landauvor-ticesintheselfdualregime,inrecentyearsmuchprogresshasalsobeenmadeawayfromtheselfdualregime(i.e.,λ=1in(1.2.10)).Infact,amuchbetterunderstand-ingofGinzburg–Landauvortexconfigurationsnowexistsindimensiond=2andd=3,alsoinrelationtotheirdynamicalproperties.Inthisrespect,seeforexam-ple:[AM],[AB],[ABG],[JMS],[BeR],[JiR],[ABP],[BPT],[BBH],[BBO],[BOS],[BR],[CHO],[CRS],[DGP],[E],[J1],[J2],[JS1],[JS2],[Lin1],[Lin2],[Lin3],[LR1],[LR2],[LR3],[MSZ],[PiR],[Riv],[RuS],[Sa],[SS1],[SS2],[SS3],[SS4],[SS5],[SS6],[SS7],[Se1],[Se2],[Se3],and[Spi].SucharemarkableunderstandingoftheGinzburg–Landaumodelhasalsopromptedtoundertakeasimilarapproachtothe6th-orderChern–Simonsmodelawayfromtheselfdualregime;contributionsinthisdirectioncanbefoundin[HaK],[KS1],and[KS2].1.3Non-abeliangaugefieldtheories1.3.1PreliminariesInthissectionwediscussexamplesofgaugefieldtheoriesdescribingphysicalinter-actionsotherthantheelectromagneticonestreatedintheprevioussection.Suchtheoriesareformulatedwithinthemathematicalframeworkof(non-abelian)gaugetheoryandarespecifiedaccordingto(arepresentationof)anassignedgaugegroupG.ThegaugegroupGisgivenbyarealLiegroup,usuallycompactandcon-nected,whichactsoverafinite-dimensional(realorcomplex)linearspaceL.Thecorrespondingrepresentation,ρ:G→Aut(L),(1.3.1)willbeusedtodescribetheinternal(local)symmetriesrelativetothetheory.Forthisreason,weshallrefertoLastheinternalsymmetryspace.Bytheusualoperationof“derivation,”wecanexchangeinformationbetweenthegroupGandits(real)LiealgebraG.Inthisway,ρin(1.3.1)inducesarepresentationofGonLwhichweshalldenoteinthesameway.Wereferto[Ca],[Fe],and[Hu]forthedetails.Inmostcaseswecanuse(1.3.1)toidentifyGwithanembeddedsubgroupoftheLiegroupof(square)realinvertiblematrices:GL(n,R)={n×nrealmatrixA:detA=0}.(1.3.2)So,formostpurposes,itisconvenienttothinkofGasamatrixgroup.Inthisrespect,notethatGcouldberepresentedalsobycomplexsquarematrices,namely,elementsofthegroup:GL(n,C)={A:n×ninvertiblecomplexmatrix};(1.3.3)butinthiscaseitisunderstoodthatG(beingarealgroup)isconsideredasubgroupofGL(2n,R). 141SelfdualGaugeFieldTheories1.3.2TheadjointrepresentationandsomeexamplesInthissection,wedescribetheadjointrepresentationforaLiegroupG.Thiswillhelpestablishideasandwillgiveusachancetoreviewarelevantgrouprepresentationforthephysicalapplications.Moreprecisely,theadjointrepresentationconcernsthesituationwhereL=G;namely,thelinear(symmetry)spacecoincideswiththeLiealgebraofG,andAd:G−→AutG(1.3.4)isdefinedasfollows.Lete∈GbetheunitelementofG,i.e.,ge=g=eg∀g∈G.Andforg∈G,definetheinnerautomorphismonGasIg:G−→Ga−→gag−1.ThenAd(g)=d(Ig)|e∈AutG,(1.3.5)whereddenotestheusualdifferentiationofsmoothfunctionsonmanifolds.TheinducedadjointrepresentationovertheLiealgebraG,ad:G−→EndGwithad=d(Ad)|e,(1.3.6)canbeshowntoactsimplybytheLiebracket[,]onG,namely,ad(A)=[A,·](1.3.7)(see[Fe]).The(real)adjointrepresentationintroducedabovecanbeextendedinanaturalwayoverthecomplexificationofthe(real)LiealgebraGasGC=G⊗RC,(1.3.8)whichdefinesacomplexLiealgebraequippedwiththeinducedLiebracket.GCpre-servesthestructuralpropertiesofGandwemayconsidertheextendedmapsAd:G−→Aut(GC)(1.3.9)andad:GC−→End(GC)(1.3.10)A−→[A,·],whereforg∈G,therestrictionsAd(g)|Gandad|Grecover,respectively,theadjointrepresentation(1.3.5)and(1.3.7). 1.3Non-abeliangaugefieldtheories15Weshallconsiderbothrealandcomplexadjointrepresentations.Inliterature,thecomplexadjointrepresentationisalsoreferredtoastheconjugaterepresentationofG.Inthecontextoftheadjointrepresentation,itisinterestingtoconsidertheKillingformk:GC×GC−→C,givenbythesymmetricbilinearformdefinedask(A,B)=tr(ad(A)ad(B)),(1.3.11)whereA,B∈GCandtrdenotethetraceofelementsinEnd(GC).TheKillingformisinvariantundertheadjointtransformationof(1.3.9)(andhence(1.3.5)),inthesensethat,forA,B∈GC:k(Ad(g)(A),Ad(g)(B))=k(A,B),∀g∈G;(1.3.12)andk|G×G∈R.Forlateruse,werecordtheidentityk([A,B],C)=k(A,[B,C]),(1.3.13)whichfollowsfromelementarypropertiesofLiebracketsandtraces.NotethattheKillingformvanishesidenticallywhenGisabelian,i.e.,everyelementofG(andthusofGC)commute:[A,B]=0,∀A,B∈GC.Ontheotherhand,itisalsoofinteresttoanalyzethesituationwherekisnon-degenerate,inwhichcasetherelativegroupissaidtobeasemisimplegroup.Itispossibletoshowthat(see[Fe]):Theorem1.3.1IfGisconnectedandsemisimple,thenGiscompactifandonlyifk|G×Gisnegativedefinite.InthecontextofTheorem1.3.1,theKillingformisusedtoprovideGwithaRiemannianstructure.Inthesequel,weshallexploreitsroleintheCartan–WeyldecompositionofGC.Next,letusseehowtheadjointrepresentationoperatesoveramatrixgroup.Forthispurpose,recallthattheLiealgebraassociatedtoGL(n,R)isgivenbygl(n,R)={n×nrealmatrices},(1.3.14)whosecomplexification(gl(n,R))C=gl(n,C)={n×ncomplexmatrices}(1.3.15)justcorrespondstotheLiealgebraofthecomplexLiegroupGL(n,C).IfGisa(embedded)subgroupofGL(n,R),thenitisnotdifficulttoseethat:ifg∈G⊂GL(n,R)andA∈G⊂gl(n,R)(orA∈GC⊂gl(n,C)),thenAd(g)A=gAg−1∈G(orGC),(1.3.16)whereweusetheusualmatrixmultiplication. 161SelfdualGaugeFieldTheoriesInfact,throughanabuseofnotation,theexpression(1.3.16)canbeadoptedtodenote,ingeneral,the(realorcomplex)adjointrepresentationforanygroup.Nextwerecallsomeexamplesofmatrixgroupsthatwillenterasgaugegroupsinthegaugefieldtheoriesdiscussedbelow.SpecialOrthogonalGroup:SO(n)=A∈GL(n,R):AtA=Id,detA=1(1.3.17)wheretAisthetransposeofA,anditsassociatedLiealgebraisso(n)=A∈gl(n,R):A=−tA.(1.3.18)SpecialUnitaryGroup:ThecomplexcounterpartofSO(n)isgivenby,SU(n)=A∈GL(n,C):AA†=Id,detA=1(1.3.19)whereA†=tAistheHermitianconjugateofA,anditsassociatedLiealgebraissu(n)=A∈gl(n,C):A=−A†,trA=0.(1.3.20)Inparticularnotethatdim2−1,(1.3.21)Rsu(n)=nandthecomplexificationofsu(n)isgivenby(su(n))C=sl(n,C)={A∈gl(n,C):trA=0}.(1.3.22)WerecallthatSU(n)definesacompact,connectedsubgroupoftheunitarygroupU(n)=A∈GL(n,C):AA†=Id(1.3.23)andalsodefinesasemisimplegroupforn≥2(cf.[Ca],[Fe]).Whileforn=1,U(1)={z∈C:|z|=1}definesanabeliancompactgroupwhichwecanidentifywithSO(2).Infact,everyelementofSO(2)canbedescribedbyapairofcomplexnumbers{z,ω}suchthatω=izand|z|=1.WethusseethattheprojectionmapfromC2intoCdefinesaLiegrouphomeomorphismbetweenSO(2)andU(1).NotethatU(1)actsonCbymultiplicationbyaunitarycomplexnumber.InconcludingthissectionwereturntotheroleofdifferentiationasameanstotransferinformationfromGover(itsinfinitesimalexpression)G,aswehaveseenalreadyfrom(1.3.5)to(1.3.6).Asiswell-known,thisproceduremaybereversedbymeansoftheexponentialmap.ForamatrixgroupG⊂GL(n,R)(orG⊂GL(n,C))withLiealgebraG⊂gl(n,R)(orG⊂gl(2n,R)),thisissimplydescribedbytheexponentialofamatrix:+∞AkA∈G−→eA=∈G.(1.3.24)k!k=1 1.3Non-abeliangaugefieldtheories171.3.3GaugefieldtheoriesWereturntothedescriptionofagaugetheoryovera(real)LiegroupG,actingoverthesymmetryspaceLwiththecorrespondingrepresentationρin(1.3.1).FromnowonweshallassumeGtobecompact.Inthissituation,itisalwayspossibletodefineaninnerproductoverLandGinvariantunderthetransformationofρ(g)andAd(g),respectively,∀g∈G(cf.[Ca]).Wedenoteitby(·,·)regardlessofwhetheritpertainstoLorG,sothatnoconfusionshouldarise.Andwelet|·|denotetheassociatednorm.AgaugefieldtheorywithgaugegroupGisformulatedintermsofthefollowingdynamicalvariables:thegaugepotentialA:aconnectionovertheprincipalbundlePwithstructuregroupG,theHiggs(matter)fieldφ:asmoothsectionoftheassociatedbundleEwithfiberL;(cf.[JT],[Tra],and[Jo]).Forthepurposeofthesenotes,itsufficestolimitourattentiontogaugetheorieswhereboththeprincipleandtheassociatedbundlearetrivial;althoughitshouldbementionedthatnon-trivialbundlesdooccurinphysicalliterature.Morepreciselywetake,P=R1+d×G,E=R1+d×L,with(R1+d,g)theMinkowskispaceandg=diag(1,−1,...,−1).Hence,thepotentialfieldAisthegloballydefinedG-valued1-formα,AA=Aαdxα=Aα(x)∈Gα=0,1,...,d,(1.3.25)andtheHiggsfieldφisthesmoothL-valuedfunctionφ:R1+d−→L.(1.3.26)Inturn,ifweletDAdenotethe(exterior)covariantderivativeassociatedtotheconnec-tionAactingonG-valuedforms,thenweobtainthegaugefieldFAasthecurvature2-formcorrespondingtoA:1αβFA=DAA=dA+A∧A=Fαβdx∧dx,(1.3.27)2withFαβ=∂αAβ−∂βAα+Aα,Aβ,(1.3.28)α,β=0,1,...,d.Bymeansofthe(induced)representationofρonG,wecanalsoconsiderthecovariantderivativeDAactingoverL-valuedformsasfollows:DAω=dω+ρ(A)∧ω. 181SelfdualGaugeFieldTheoriesInthisway,wecan(weakly)coupletheHiggsfieldφtothepotentialAasfollows:αwithDDAφ=Dαφdxαφ=∂αφ+ρ(Aα)φ,(1.3.29)α=0,1,...,d.ObservethatwhenwerepresentGaccordingtothe(realorcomplex)adjointrep-resentation,thecomponentsofthecovariantderivativeofφreducetotheexpressionDαφ=∂αφ+[Aα,φ],(1.3.30)α=0,1,...,d.A(relativistic)gaugefieldtheory(innormalizedunits)isformulatedbymeansofaLagrangeandensityoftheform11Fαβ+αφ−V,(1.3.31)L(A,φ)=−αβ,FDαφ,D42withscalarpotentialVtypicallyassignedwithdependenceon|φ|2=(φ,φ).TheinternalsymmetriesofthetheoryarenowexpressedbytheinvarianceofLunderthegaugetransformationsA−→A−1,(1.3.32)g=Ad(g)A+gdgφ−→φg=ρ(g)φ,(1.3.33)foranygivensmoothgaugemapg:R1+d−→G.(1.3.34)Indeed,itisnotdifficulttoverifythefollowing(covariant)transformationrules:FAg=Ad(g)FA,DAgφg=ρ(g)DAφ.(1.3.35)Consequently,theinvarianceoftheinnerproduct(relativetoLorG)immediatelygivestheinvarianceofLunderthetransformations(1.3.32)and(1.3.33).Tofamiliarizeourselveswithsuchaframework,letusseehowtorecasttheelec-tromagnetictheorydiscussedintheprevioussectionwithinthisformalism.Weseethat,forthispurpose,weneedtospecifythegaugegroupG,asgivenbythe(abelian)groupofrotationsinR2,andtheinternalsymmetryspace,asgivenbythecomplexline.Namely,wetakeG=U(1)≡SO(2)andL=C,(1.3.36)andconsiderCequippedwiththestandardinnerproduct(z,w)=zw¯∀z,w∈C.(1.3.37)WeknowthatU(1)definesacompact(topologicallyS1)abelianLiegroup,actingonCasmultiplicationbyaunitarycomplexnumber.TheassociatedLiealgebraGistheimaginaryaxis,whichisrepresentedasG=−iRandwhichactsonCbymultiplication. 1.3Non-abeliangaugefieldtheories19Consequently,forthematterandpotentialfields,wefindthattheexpressions(1.2.2)and(1.2.3).Furthermore,inaccordancewith(1.3.27),(1.3.28)and(1.3.29)—thecorrespondinggauge(Maxwell)fieldandcovariantderivativeofφ—taketheformsof(1.2.4),(1.2.5)and(1.2.6),respectively.Inaddition,agaugetransformationgoverU(1)isassignedsimplybyasmoothfunctionω:R1+d−→Rasfollows:g:R1+d−→U(1)x−→eiω(x).Withthisinformation,wecanrevisittheMaxwell–Higgstheorydiscussedinthepre-vioussectiontofitwithinthegaugefieldformalismillustratedabove.Moregeneral(non-abelian)gaugefieldtheoriesmaybeformulatedwhenwereplaceU(1)withother(matrix)groups.Thus,whileU(1)pertainstoelectromag-netism,thegroupSU(2)isinvolvedintheformulationoftheYang–Mills–Higgstheoryofweakinteractions,whileSU(3)istheappropriategaugegrouptode-scribestronginteractions.InthegroupSU(5)liesthehopefordescribingauniversalunifiedtheorybeyondthealreadycelebratedelectroweaktheoryofGlashow–Salam–Weinbergwhichformulatesaunified(SU(2)×U(1))-gaugefieldtheoryforelectro-magneticandweakinteractions.Weshallpresentamoredetaileddiscussionofnon-abeliangaugefieldtheoriesinthecontextofthe(realorcomplex)adjointrepresentation,whereboththepotentialandthematterfieldsareexpressedonthesamebasisofthegaugealgebra(realorcom-plex).Asisalreadyapparentfromtheexpressionofthecovariantderivative(1.3.30),itisimportanttoconsideronG(orGC),abasisthatsatisfiesthemostconvenientcommutatorrelations.WewillreachthisgoalbymeansoftheCartan–Weylbasisdecomposition(cf.[Ca],and[Hu])whichwediscussbrieflyinthefollowingsection;wewilldescribethemostrelevantfeatures,especiallyinthecontextofsemisimpleLiegroups(e.g.,SU(n)).1.3.4TheCartan–Weylgenerators:basicsInthissectionwegiveabriefaccountontheCartan–Weyldecompositionofacom-plex(finite-dimensional)semisimpleLiealgebra{L,[,]}accordingtothe(complex)adjointrepresentation;wereferto[Ca]and[Hu]fordetails.Tothispurpose,werecallthatthereexistsanelementA∈L,suchthatthelinearmapad(A):L−→L(1.3.38)X−→[A,X]admitsonlyzeroasamultipledegenerateeigenvalue,whilethenumberofremainingnon-zeroeigenvaluesismaximal.Suchnon-zeroeigenvaluesarecalledtherootsofLandnecessarilymustbesimpleeigenvalues.Theeigenspaceofthezeroeigenvalue(i.e.,Kerad(A))containsA,andbyvirtueoftheJacobiidentity[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0,(1.3.39) 201SelfdualGaugeFieldTheoriesdefinesasubalgebraofL,whichcanbeshowntobeabelian(i.e.,eachpairofitselementscommuteundertheLiebracketoperation).Actually,theeigenspacecoincideswiththemaximalabeliansubalgebraofL,andthusisindependentofthechoiceoftheelementA.SuchamaximalabeliansubalgebraofLisknownastheCartansubalgebra,anditsdimensionrdefinestherankofL.Denotingby{Ha}a=1,...,rabasisfortheCartansubalgebra,wehave:[Ha,Hb]=0∀a,b=1,...,r.(1.3.40)SinceAbelongstotheCartansubalgebrawemaywrite:A=γaHa.(1.3.41)Now,letλbearootofLanddenotebyEλacorrespondingeigenvector,calledarootoperator.Then[A,Eλ]=λEλ.(1.3.42)BymeansoftheJacobiidentity(1.3.39),wefindthat[A,[Ha,Eλ]]=−[Ha,[Eλ,A]]−[Eλ,[A,Ha]]=λ[Ha,Eλ]∀a=1,...,rwherewehaveused(1.3.40)and(1.3.41).Thatis,[Ha,Eλ]isalsoaneigenvectorcorrespondingtothe(simple)eigenvalueλ,andconsequently[Ha,Eλ]=αaEλ∀a=1,...,r.(1.3.43)Inthisway,toeachrootλwecanassociateavectorα=(α1,...,αr)∈Crknownasarootvector.Notethatarootvectorisnon-zero,i.e.,α=0.Itispossibletocheckthattherootvectorαisuniquelydeterminedbyλ,inthesensethatdifferentrootsgiverisetodifferentrootvectors.By(1.3.41)thefollowingrelationholds:aαλ=γa.(1.3.44)Thus,ifwedenoteby⊂Cr{0}thesetofrootvectors,wefindaone-to-onecorrespondencebetweenandthesetofrootsofL.ForthisreasonwecanreplaceEλwithEαtoidentifythecorrespondingrootop-erator.Furthermore{Ha,Eα}definesabasisforL,knownastheCartan–Weylbasis,andforeachX∈Lweobtainthefollowingrootspacedecomposition:ryaHX=a+yαEα.(1.3.45)a=1α∈Nextnoticethat,ifα=(αa)a=1,...,randβ=(βa)a=1,...,r∈aretworootvectorsof,withλα=γaαaandλβ=γaβathecorrespondingroots,thenbymeansoftheJacobiidentity(1.3.39),wecanverifytheidentity:[A,[Eα,Eβ]]=(λα+λβ)[Eα,Eβ].(1.3.46) 1.3Non-abeliangaugefieldtheories21Thusweseethat,eitherEα,Eβcommuteas[Eα,Eβ]=0,(1.3.47)orλα+λβisarootofLandwemustdistinguishbetweenthefollowingcases:i)λα+λβ=0,i.e.,β=−α,wherewefindaHa[Eα,E−α]=αaforsuitableα∈C(1.3.48)ii)λα+λβ=0,i.e.,α+βisarootofAandwefind[Eα,Eβ]=cαβEα+βforsuitablecαβ∈C.(1.3.49)Asamatteroffact,whenEαandEβcommute,wecansupposethattherelations(1.3.48)and(1.3.49)arestillvalidbytakingαa=0orcαβ=0,respectively.Inparticular,fromidentity(1.3.49)weseethatinordertorepresentLwedonotneedtoincludeEα+β,amongitsrootoperatorswhenα+β=0;thiseigenvectormayberecoveredviathebracketoperationfromEαandEβ,providedtheydonotcommute.Therefore,itbecomesaninterestingproblemtodeterminetheminimalsetofrootvectors,knownasthesimplerootvectors,whichgeneratethewholesetbysummingoveritselements.Inthisway,theLiealgebraLwouldbecompletelyrepresentedbythegeneratorsoftheCartansubalgebraandthesimplerootstepoperators.ThesetofsimplerootstepoperatorsisparticularlynicetodescribeforsemisimpleLiealgebraforwhichwecanderivedirectinformationoutoftheKillingform(1.3.11)intherootspacedecomposition.Tothispurpose,let{Xl}beabasisforLwithstructuralconstantsCml,ndefinedbytherelation:lX[Xm,Xn]=Cm,nl.(1.3.50)Then,theKillingformkmaybecompletelyexpressedintermsofthesymmetricsquarematrixg=gm,n,withglCν,(1.3.51)m,n=Cm,νnlaswefindthatagbllk(X,Y)=xa,by,forX=xXlandY=yXl.Therefore,ifLissemisimple,theng=gm,nisinvertible(i.e.,detg=0).Fur-thermore,ifLcorrespondstothecomplexificationofarealLiealgebrarelativetoacompact,connectedsemisimplegroup,thenbyTheorem1.3.1,wemayevenconcludethatgisnegativedefinite.Letusseehowtousethisinformationwhenwefix{Ha,Eα,a=1,...,r,α∈}asabasisforLaccordingtotherootspacedecomposition. 221SelfdualGaugeFieldTheoriesAsaconsequenceofthecommutatorrelations(1.3.40),(1.3.43),(1.3.48),and(1.3.49),weobtainthestructuralconstantstobegivenasfollows:βlCab=0=Cabβ∈ ,a,b,l∈{1,...,r}(1.3.52)Cβ=αβlaαaδα,Caα=0α,β∈ ,a,b,l∈{1,...,r}(1.3.53)ββcαµδα+µifα+µ=0l0α+µ=0Cαµ=Cαµ=aa(1.3.54)0otherwiseαδα+µ=0lα,β,µ∈ ,l∈{1,...,r}.Byvirtueof(1.3.52),(1.3.53),and(1.3.54),wefindthatthematrixgisformedbytwoblocks:{gab}a,b=1,...,rgivenbyitsrestrictionovertheCartansubalgebra,gab=k(Ha,Hb);{gαβ}α,β∈givenbyitsrestrictionovertherootspaceoperatorsgαβ=k(Eα,Eβ),whilegcontainszeroselsewhere.Infact,fora∈{1,...,r}andα∈wefindlblβνβνbgαa=CαbCal+CαβCal+CαβCaν+CαbCaν=0=gaα.(1.3.55)Ontheotherhand,whenα,µ∈wehavelblβνβνbgαµ=CαbCµl+CαβCµl+CαβCµν+CαbCµν=gµα.Weseethatnecessarily−α∈;otherwise,wewouldgetgαµ=0forallµ∈,andinlightof(1.3.55),wewouldcontradictthenon-degeneracyofg.Thus,bytheanalysisoftheblock{gαµ}wehavediscoveredafirstimportantprop-ertyofsemisimpleLiealgebras;namely,ifα∈ ,then−α∈ .(1.3.56)Next,weconsidertheblock{gab}a,b=1,...,rrelativetotherestrictionofgontheCartansubalgebra.First,notethat{gab}definesanon-degenerater×rmatrix.Inviewof(1.3.53)and(1.3.48),wecomputeαak(Eα,E−α)=k([Ha,Eα],E−α)=k(Ha,[Eα,E−α])bk(H=αa,Hb)=αbgab,(1.3.57)wherewehaveused(1.3.13). 1.3Non-abeliangaugefieldtheories23Therefore,ifwenormalizetherootoperatorstosatisfyk(Eα,E−α)=1,thenfrom(1.3.57)wededucea=gabααb,(1.3.58)where,asusual,(gab)denotestheinversematrixof(gab).Byvirtueof(1.3.58),itseemsappropriatetointroducethefollowinginnerproductovertheset:aβab(α,β)=αa=βagαb.(1.3.59)Furthermore,intermsofthestructuralconstantsweseethatglCm+ClCα+CβCm+CβCαab=Camblaαblambβaαbββα=CaαCbβ=α∈αaαb,andthenon-degeneracyof{gab}impliesthatthesetofrootvectorsmustspanthewholespaceCr.Suchapropertyindicatesthatthesetofsimplerootsmustcontainexactlyrinde-pendentvectors.Thisresultcanbemaderigorous(cf.[Ca]and[Hu])asitispossibletoshowthat,forasemisimpleLiealgebrathesetofsimplerootscontainsexactlyrindependentvectorsandanyotherrootvectorcanbeobtainedasasumofsimplerootvectorswithintegercoefficientsallhavingthesamesign(either≥0or≤0).Moreprecisely,=α(a)a=1,...,rand∀α∈:(a)+−(1.3.60)α=naαwithna∈Z∀a,orna∈Z∀a.Therootoperatorcorrespondingtothesimplerootα(a)willbedenotedbyEa,a=1,...,randcalledasimplerootstepoperator.Notethat,accordingto(1.3.60),fora=bwehave[Ea,E−b]=0,asα(a)−α(b)∈/.Infact,accordingtoChevalley’snormalizationconditions(cf.[Ca],[Hu],and[Chev]),itisalwayspossibletoarrangetheCartansubalgebragenerators{Ha}a=1,...,randthesimplerootstepoperators{Ea}a=1,...,rtosatisfythefollowingcommutatorandtracerelations:[Ha,Hb]=0(1.3.61)[Ea,E−b]=δaHa(1.3.62)b[Ha,E±b]=±KabE±b(1.3.63)tr(EaE−b)=δa,tr(HaHb)=Kab,tr(HaE±b)=0(1.3.64)ba,b=1,...,r, 241SelfdualGaugeFieldTheorieswherethecoefficientsα(a),α(b)Kab=2(1.3.65)α(a),α(a)formanr×rmatrix,knownastheCartanmatrix.TheabovedecompositioncanbenicelyillustratedforthegroupSU(n),wherethecorrespondingoperatorsHaandEaaredeterminedexplicitly.WestartwiththeparticularlysimplecaseofSU(2),wheretheassociatedLiealgebrasu(2)=A∈gl(2,C):A†=−A,trA=0admits(real)dimension3.Itscomplexificationsl(2,C)={A∈gl(2,C):trA=0}alsoadmits(complex)dimension3.Weeasilydetermineabasisforsl(2,C),asgivenby:100100T1=,T2=,T3=.0−10010†ObservethatT3=T,andthefollowingcommutatorrelationshold:2[T1,T2]=2T2,[T1,T3]=−2T3,(1.3.66)[T2,T3]=T1.(1.3.67)Hence,nopairoflinearlyindependentelementsofsl(2,C)cancommute.Therefore,thecorrespondingCartansubalgebraisone-dimensional(i.e.,r=1),andinviewof(1.3.66)wearriveattheconclusionthatH1=T1,E1=T2givetheCartansubalgebrageneratorandthesimplerootstepoperator,respectively,†whileE−1=E=T3.ItcanbecheckedbydirectinspectionthatalltheChevallery’s1relations(1.3.61)–(1.3.64)areverifiedsinceK=K11=2inthiscase.Toobtainabasisforsu(2),wesimplyhavetouselinear(complex)combinationsofH1,E1andE−1toobtainthreelinearlyindependentanti-Hermitiantracelessmatrices.Easilyweguessthemtobe:iH1,E1−E−1,i(E1+E−1).Infact,itismoreconvenienttotakeasabasisforsu(2)thefollowing:111X1=(E1+E−1),X2=−(E1−E−1),X3=H1.(1.3.68)2i22iThisbasissatisfies[Xa,Xb]=εabcXc,(1.3.69) 1.3Non-abeliangaugefieldtheories25andisrelatedtothe(Hermitian)Paulimatrices:010−iσ1==E1+E−1,σ2==i(E−1−E1),(1.3.70)10i010σ3==H1(1.3.71)0−11σviatheidentityXa=a,a=1,2,3.2iThisfirstexamplegivesahintonhowtoproceedmoregenerallyforthegroupSU(n).WeintroducethematricesTabdefinedbythecondition:(Tabab)jk=δjδk(1.3.72)a,b=1,...,n.SetHa=Taa−Ta+1,a+1,a=1,...,n−1,(1.3.73)andnotethatsuchmatricesarediagonalandhencecommuteas[Ha,Hb]=0,a,b=1,...,n−1.†Inaddition,T=TbaandTab∈sl(n,C)onlyifa=b.Infact,thesetab†Ha,Tab,T,1≤a0definestheChern–Simonscouplingpara-meterwhosestrengthmustbecounterbalancedbythestrengthofthescalarpotentialV=V(φ,φ†),inorderfor(1.3.99)tosupportaselfdualstructure.Tothispurpose,Dunnein[D1](seealso[D2])proposedthefollowinggauge-invariantscalarpotential:

†1†2†2V=trφ,φ,φ−νφφ,φφ−νφ(1.3.101)k212=φ,φ†,φ−ν2φ,k2withthesymmetry-breakingmass-scaleparameterν2.AlthoughthepotentialVin(1.3.101)mightappearunusualatfirstsight,weun-coveritsfamiliarnaturebylookingatitoverthesubspaceofsl(n,C),generatedbythesimplerootstepoperatorsEa,a=1,...,n−1in(1.3.77).Infact,forφ=φaEa,withφa∈C,wefind1a2b22V=|φ(ν−Kba|φ|)|,(1.3.102)k2whichmightbetakenasthenaturalextensionoverthe(n−1)-ples{φa}oftheself-dualChern–Simonspotential(1.2.40)withcouplingprovidedbytheCartanmatrixin(1.3.79).TheEuler–Lagrangeequationsrelativeto(1.3.95)aregivenbyα∂VDαDφ=−,(1.3.103)∂φ†kµαβµεFαβ=−iJ,(1.3.104)2with
Jµ=iDµφ,φ†−φ,(Dµφ)†,(1.3.105) 321SelfdualGaugeFieldTheoriesasthecorrespondingnon-abeliancovariantlyconservedcurrent.Againwenoticeanobviousformalanalogybetween(1.3.105)anditsabeliancounterpart(1.2.35).Fromtheµ=0componentof(1.3.104)weobtainthe(non-abelian)Gausslawconstraint:1†00†F12=−([φ,Dφ]−[(Dφ),φ]).(1.3.106)kAsfortheabeliancase,wecanuse(1.3.106)todeducethefollowingexpressionforthegauge-invariantpartoftheenergydensity:
E=trDαφ)†+V(1.3.107)αφ(Dwhereagain,theChern–Simonstermdoesnotcontributeto(1.3.107).TorevealhowthechoiceofVin(1.3.101)impliesaselfdualstructure,werecordthefollowingidentityasthenon-abeliancounterpartof(1.2.38):jk222†ε|D1φ|+|D2φ|=|D±φ|±itrF12φ,φ±∂jQk.(1.3.108)2HereD±φ=D1φ±iD2φ;(1.3.109)and
µ†µµ†Q=itrφD−Dφφ(1.3.110)identifiestheabeliancurrentrelativetotheU(1)-invarianceofthetheory.WiththisinformationandwiththechoiceofthescalarpotentialVin(1.3.101),wecanproceedanalogouslytotheabeliansituationandarriveatthefollowingexpressionfortheenergydensity:12εjk2+|D2†φ,φ†2E=|D0φ|±φ|±itrF12φ,φ+,φ−νφ±∂jQkk22
2=D0φ∓iφ,φ†,φ−ν2φ+|D±φ|2(1.3.111)k2
jk톆ε±itrD0φφ−φ(D0φ)±∂jQk.k2Thus,byusingtheelementarypropertiesoftracesandthecomponentQ0oftheabeliancurrent(1.3.110),wefind
22jkE=D0φ∓iφ,φ†,φ−ν2φ+|D±φ|2±νQ0±ε∂jQk.(1.3.112)kk2Therefore,withinappropriateboundaryconditionsthatallowustoneglectthetotalspatialdivergencetermin(1.3.112),wecansaturatetheenergylowerboundthroughthesolutionsofthefollowing(first-order)equations: 1.3Non-abeliangaugefieldtheories33D±φ=0(1.3.113)Diφ,φ†,φ−ν2φ,(1.3.114)0φ=±ktobesatisfiedinadditiontotheGausslawconstraint(1.3.106).Inotherwords,(1.3.106),(1.3.113),and(1.3.114)representtheselfdualequationscorrespondingtothenon-abelian(pure)Chern–Simonsmodel(1.3.99)and(1.3.101).Inordertoiden-tifythestaticselfdualsolutions(i.e.,solutionsindependentofthex0-variable),itisconvenienttosubstitute(1.3.114)into(1.3.106)andrewritethe(static)selfdualequa-tionsintheformD±φ=02i2††F12=∓νφ−φ,φ,φ,φ;(1.3.115)k2andthenuse(1.3.114)todetermineonlytheA0-componentofthepotentialfield.Incidentally,noticethattheselfdualequations(1.3.115)aretrivializedintherealadjointrepresenationwhereφ∈su(n),andsoφ†=−φ.Indeedinthiscase,thesecondequationin(1.3.115)leadstoF12=0,while(1.3.114)canonlyresultinthetrivialsolutionφ=0sinceitsleft-handsideisHermitianwhileitsright-handsideisanti-Hermitian.SotheselfdualChern–Simonstheorypresentedhereisoftrueinterestinthecon-jugaterepresentation.Theresulting(non-abelian)selfdualequations(1.3.114)and(1.3.115)aremuchmoredifficulttohandleincomparisonwiththeirabeliancoun-terpart(1.2.45),inspiteoftheiranalogiesata“formal”level.Infact,concerning(1.3.114)and(1.3.115)rigorousanalyticalresultshavebeenobtainedonlyundertheansatzthateachcomponentAαofthepotentialfieldtakesval-uesontheCartansubalgebraofsu(n),whilethematterfieldφbelongstothesubspaceofsl(n,C)generatedbythesimplerootstepoperators.Observethatthecommutatorrelations(1.3.61)–(1.3.64),validfortheCartan–Weylgenerators(1.3.77),provecon-sistencyoftheselfdualequationsundersuchrestrictions.Moreprecisely,weletaHaAα=−iAαaAα∈R,α=0,1,2(1.3.116)aEaφ=φaφ∈C,(1.3.117)a=1,...,n−1;andseethat
Daaaaabaαφ=DαφEawithDαφ=∂αφ−iAαKbaφ,(1.3.118)whereKbaarethecoefficientsoftheCartanmatrix.Furthermore,Faaaaαβ=−iFαβHawithFαβ=∂αAβ−∂βAα,(1.3.119)wherenocommutatorappears,as{Ha,a=1,...,n−1}commute;and
D††aJ0=i0φ,φ−φ,(D0φ)=J0Ha,(1.3.120) 341SelfdualGaugeFieldTheorieswith
Ja=iφaDaφa−Daφaφa.(1.3.121)000Therefore,settingDaφa=Daφa±iDaφa,(1.3.122)±12wecanexpress,bystraighforwardcalculations,theselfdualequations(1.3.115)com-ponentwiseasDaφa=0,±
a22b2a2(1.3.123)F12=±2ν−|φ|Kba|φ|;kwhiletheGausslawconstraint(1.3.106)becomesa1aF12=J0,(1.3.124)ka=1,...,n−1.Alsonoticethat,fortheabeliancurrentdensity,wehaven−1Q0=Ja.(1.3.125)0a=1Inotherwords,asolutionto(1.3.123)and(1.3.124)giverise(via(1.3.116)and(1.3.117))tothesolutionsfortheselfdualequations(1.3.114)and(1.3.115),whoseenergydensityreduceston−1E=±ν2Fa+spatialdivergenceterms.(1.3.126)12a=1Itisnotsurprisingthatforn=2,thereducedSU(2)-selfdualequationsabovejustco-incidewiththeselfdualequationsoftheabelianU(1)-Chern–Simonsmodel(1.2.41).Ontheotherhand,forn≥3,theselfdualequations(1.3.123)and(1.3.124)rep-resentasystemofselfdualabelian-typeequationscoupledthroughtheCartanmatrix(1.3.79).Thisfactisalsoexpressedbythenatureofthegaugeinvariancepropertiesofequations(1.3.123)and(1.3.124),givenbythetransformationrules:a−→Aa+∂Aαααωa,(1.3.127)aiωbKbaaφ−→eφ.(1.3.128)Wereferto[D1]and[Y1]foradiscussionofothernon-abelianChern–Simonsmodelsconcerningnon-relativisticsettings. 1.4Selfdualityintheelectroweaktheory351.4SelfdualityintheelectroweaktheoryAsalastexample,wepresentAmbjornOlesen’sapproach(cf.[AO1],[AO2],and[AO3])todescribeaselfdualstructureforthecelebratedelectroweaktheoryofGlashow–Salam–Weinberg(cf.[La])ofunifiedelectromagneticandweakforces.Theelectroweaktheoryisarelativistic(SU(2)×U(1))-gaugefieldtheorythatwearegoingtoconsiderovertheMinkowskispace(R1+3,g)withmetrictensorg=diag(1,−1,−1,−1).ThegaugegroupG=SU(2)×U(1)actsoverC2(theinternalsymmetryspace)accordingtotherepresentationρ:SU(2)×U(1)−→Aut(C2).(1.4.1)Wearegoingtodescribethisrepresentationintermsofthematrices:1ta=σa,a=1,2,3,(1.4.2)2whereσaisthe2×2Paulimatrixanda=1,2,3,isdefinedin(1.3.70)and(1.3.71).Tothispurpose,notethatbyvirtueof(1.3.68)and(1.3.69),weknowthat{−ita}a=1,2,3definesabasisforsu(2),andthereholds:[ta,tb]=iεabctc,(1.4.3)Trta,(1.4.4)atb=δbwhereTr=2tristhenormalizedtrace.Hence,wecanusetheexponentialmap(1.3.24)toexpresseveryelementofSU(2)asfollows:−iλataae∈SU(2),λ∈R.(1.4.5)Moreover,letting110t0=,(1.4.6)201wemayextendthenotationin(1.4.5)overthegroupU(1),byrepresentingeveryelementofU(1)asa2×2complexmatrixintheform:e−iξt0∈U(1),ξ∈R.(1.4.7)Then,forh=(e−iλata,e−iξt0)∈G=SU(2)×U(1)andϕ=ϕ1∈C2,theϕ2representationρofGonC2isdefinedas−igλata−ig∗ξt02ρ(h)ϕ=eϕ∈C,(1.4.8)withgandg∗thecouplingconstantsrelativetothegroupSU(2)andU(1),respectively. 361SelfdualGaugeFieldTheoriesInthissetting,thepotentialfieldisexpressedbythepairA=−igAαdxαwithAα=Aaαta,α(1.4.9)B=−ig∗Bαt0dx,whereAaα,a=1,2,3andBαaresmoothrealfunctionsoverR1+d,α=0,1,2,3.Thecorrespondinggaugefieldsaregivenbytheexpressions:iα∧dxβ,withF(1.4.10)FA=−gFαβdxαβ=∂αAβ−∂βAα+igAα,Aβ2iαβFB=−g∗Gαβt0dx∧dx,withGαβ=∂αBβ−∂βBα.(1.4.11)2ConcerningtheHiggsmatterfieldφ,itissimplyexpressedbyaC2-valuedsmoothfunction1+32ϕ1φ:R−→C,φ=;(1.4.12)ϕ2anditisweaklycoupledtothepotentialfieldbymeansofthecovariantderivativeatDαφ=∂αφ−igAαaφ−ig∗Bαt0φ,α=0,1,2,3.(1.4.13)OverC2,weconsiderthestandardinnerproduct:†ϕ1ψ12(,)==¯ϕ1ψ1+¯ϕ2ψ2,for=,=∈C.(1.4.14)ϕ2ψ2While,withtheidentification(1.4.7),wemayusetheusualinnerproduct(1.3.83)fortheelementsofthegaugealgebra.Inthisway,thebosonicsectoroftheelectroweaktheoryisformulatedaccordingtotheLagrangeandensity1
†1†2TrFFαβ2αβαL(A,B,ϕ)=−gαβ−g∗GαβG+Dφ(Dαφ)44
2−λϕ2−φ†φ,(1.4.15)0withλandϕ0twopositiveparameters.Forωa,ξ:R1+3→R,smoothfunctions(a=1,2,3),andf=eigωata∈SU(2),wecheckthattheLagrangeandensityin(1.4.15)isinvariantunderthegaugetransformations:igωata+ig∗ξt0ϕ−→eϕ,A−1−1α−→fAαf+f∂αf,(1.4.16)Bα−→Bα+∂αξ. 1.4Selfdualityintheelectroweaktheory37Byvirtueofsuchinvariance,wecanrewritethetheoryintheunitarygaugedefinedbythecondition0φ=,(1.4.17)ϕwithϕarealscalarfunction.Infact,intheunitarygauge,itisconvenienttoformulatethetheoryintermsofthenewfields’configurations—Wα,PαandZα—obtainedasthelinearcombinationsoftheoriginalfieldsasfollows:P3α=Bαcosθ+Aαsinθ,(1.4.18)Z3α=−Bαsinθ+Aαcosθ,(1.4.19)and
112Wα=√Aα+iAα,(1.4.20)2α=0,1,2,3andθ∈(0,π).Tocomprehendtheroleoftheangleθ,weobserve2thatWαandZαaremassivefieldsmediatingshort-range(weak)interactions,whilePαmediateslong-range(electromagnetic)interactions(see[La]).IntermsofPαandZα,thecovariantderivativeDαtakestheexpression:
A1t2Dα=∂α−igα1+Aαt2−iPα((gsinθ)t3+(g∗cosθ)t0)(1.4.21)−iZα((gcosθ)t3−(g∗sinθ)t0).ForPαtomediateelectromagenticinteractions,weneedtherelativecoefficientin(1.4.21)tocorrespondtothechargeoperatoreQ=e(t3+t0),whereeistheelectron’scharge.Consequentlywederive:e=g∗cosθ=gsinθ.Thatisgg∗ge=andcosθ=,(1.4.22)11g2+g∗22g2+g∗22θ∈(0,π).2TheangleθisknownastheWeinberg(mixing)angle,andfromnowonitwillbefixedaccordingto(1.4.22).Inthisway,weseethat
A1t2Dα=∂α−igα1+Aαt2−iPαeQ−iZαeQ,(1.4.23)withQ=(cotθ)t3−(tanθ)t0.Consequently,intheunitarygaugewhereφisassignedaccordingto(1.4.27),wehave−igA1−iA2ϕDαφ=2αigα,(1.4.24)∂αϕ+Zαϕ2cosθwithϕarealscalarfunction. 381SelfdualGaugeFieldTheoriesTherefore,ifweletDα=∂α−igA3αandsetPαβ=∂αPβ−∂βPα,Zαβ=∂αZβ−∂βZα,thenbydirectinspectionwecanconfirmthattheLagrangeandensityin(1.4.15),whenexpressedintheunitarygaugevariables,takestheform:1αββα1αβ1αβL=−DαWβ−DβWαDW−DW−ZαβZ−PαβP244
12α2ααβαβ−gWWα−WWαWβWβ−igZcosθ+PsinθWαWβ21α1
2+2ϕ2WWα22α22gα+∂ϕ∂αϕ+2gϕZZα−λϕ0−ϕ.(1.4.25)24cosθToobtainselfdualelectroweakvortex-typeconfigurations,AmbjornandOlesen(see[AO1],[AO2],and[AO3])suggestedthatonetakeallmagneticexcitationasconfinedinthethirddirection,accordingtothefollowingvortexansatz:Aa=Aa=B0=B3=0,03(1.4.26)Aa=Aa(x1,x2),Bj=Bj(x1,x2),j=1,2,jja=1,2,3,andϕ=ϕ(x1,x2).(1.4.27)Inaddition,theyassumethatforacomplexfieldW,thereholdsW1=W,iW2=W;(1.4.28)thatis,√√−A2=A1=2Re(W),A2=A1=2Im(W).(1.4.29)2112Wethenfindthefollowingexpressionforthecorrespondingenergydensity212122E=|D1W+D2W|+P12+Z12−2g(Z12cosθ+P12sinθ)|W|222421222222222+2g|W|+(∂iϕ)+2gϕZj+gϕ|W|+λ(ϕ0−ϕ).4cosθNoticethat,undertheansatz(1.4.26)–(1.4.28),theinvarianceofLaccordingtothetransformations(1.4.16)issimplyexpressedbyaresidualU(1)-invariance,oftheenergyEdefinedabove,underthegaugetransformation:iξW,P1W−→ej−→Pj+∂jξe(1.4.30)Zj−→Zj,ϕ−→ϕj=1,2,foranysmoothfunctionξ=ξ(x1,x2). 1.4Selfdualityintheelectroweaktheory39Withtheaimtoattainselfduality,AmbjornandOlesenin[AO1]observedthattheaboveenergydensityEmaybewrittenintheform,1
g22+P2−2gsinθ|W|2E=|D1W+D2W|12−ϕ022sinθ1
g
2
g2+Z2−ϕ22+12−ϕ0−2gcosθ|W|ϕZj+εjk∂kϕ22cosθ2cosθg2
2g2gϕ2gϕ2+λ−2−ϕ2−4+0P02ϕ02ϕ012−Z128cosθ8sinθ2sinθ2sinθ
g2−∂kεjkZjϕ.(1.4.31)2cosθg2Therefore,forλ≥2wehave8cosθg2gϕ2gϕ2g
E≥−ϕ4+0P022012−Z12−∂kεjkZjϕ.(1.4.32)8sinθ2sinθ2sinθ2cosθInaddition,whenthegivenparameterssatisfythe“critical”conditiong2λ=,(1.4.33)8cos2θthenthelowerboundin(1.4.32)issaturatedbythesolutionsofthefollowingsystemofselfdualequations:⎧⎪⎪D1W+iD2W=0,⎪⎪⎪⎨g22P12=ϕ+2gsinθ|W|,2sinθ0(1.4.34)⎪⎪Z12=gϕ2−ϕ2+2gcosθ|W|2,⎪⎪2cosθ0⎪⎩2cosθεZj=−jk∂klogϕ,j=1,2.gAsusual,(1.4.34)representsafirst-orderfactorizationoftheEuler–Lagrangeequa-tionscorrespondingtoLin(1.4.25).Inparticular,eachsolutionof(1.4.34)givesrisetoacriticalpointforL.Notethatintheunitarygaugevariables,therealfieldϕnevervanishes.Alsoobservethatwecancombinethelasttwoequationsin(1.4.34)andobtain,
g2−logϕ=(ϕ2−ϕ2)−g2|W|2.(1.4.35)2cosθ0Expression(1.4.32)impliesthatplanarsolutionsof(1.4.34)maycarryinfiniteenergy.Therefore,intheplanarcase,theselfdualitycriterionyieldstosolutionswhichmaynotbareaphysicalmeaning.Nonetheless,weknowthatthesolutionsof(1.4.34)appearinabundanceandmaybeselectedaccordingtotheirdecaypropertyatinfinity(see[SY3]and[ChT1]).Asweshallsee,theappropriateboundaryconditionstoconsiderinthiscon-textaretheperiodicones,asintroducedby’tHooftinagauge-invariantsituation 401SelfdualGaugeFieldTheories(see[’tH2]).Thecorrespondingperiodicelectroweakvortex-likeconfigurationsareknownasW-condensates,andarephysicallyinterestingbyvirtueoftheirconnectiontotheso-calledAbrikosov’s“mixedstates”insuperconductivity.SincetheirpresencewasfirstpredictedbyAmbjornandOlesen([AO1],[AO2],and[AO3])onthegroundsofsomenumericalevidence,theexistenceofW-condensateshasbeenestablishedrigorouslyin[SY2]and[BT2].WewilldiscusstheminChapter7.Theanalysisofselfdualelectroweakvortexconfigurationsactuallycanbeextendedtoselfgravitatingelectroweakstrings;theyoccurinthetheorydescribedabove,whenwealsotakeintoaccounttheeffectofgravity.Inthissituation,themetrictensorgisnolongerfixedbutpartoftheunknownstobedeterminedaccordingtothecoupledelectroweakEinsteinequations.Again,itispossibletoshowthataselfdualregimeisattainedbytheelectroweakEinsteintheory,whenweallowthegravitationalmetrictovaryintheclass
2
2
2
2ds2=dx0−dx3−eηdx1+dx2,(1.4.36)withηtheunknownconformalfactor.Thus,asbefore,forλfixedaccordingtothecriticalcondition(1.4.33),andunderthestringansatzaccordingtowhichηisafunctiondependingonlyonthevariables(x1,x2),andwhen(1.4.26)–(1.4.29)hold,wefindthatselfgravitatingstrings(parallelalongthex3-axis)maybeobatinedbysolvingasetofselfdualequationswhichmodify(1.4.34)asfollows:⎧⎪⎪D1W+iD2W=0,⎪⎪⎪⎨g2η2P12=ϕe+2gsinθ|W|,2sinθ0(1.4.37)⎪⎪Z12=gϕ2−ϕ2eη+2gcosθ|W|2,⎪⎪2cosθ0⎪⎩2cosθεZj=−jk∂klogϕ,j=1,2,gwithηsatisfyingEinstein’sequationsthat,inthissetting,reduceto
ηgϕ2g
−=0Pϕ2−ϕ22,(1.4.38)12+0Z12+4|∇ϕ|8πGsinθcosθwhereGisthegravitationalconstant.Wereferthereaderto[Y1]forthedetails.Herewelimitourselvestoobservingthattheenergydensity,correspondingtosolutionof(1.4.37),takestheform:g2ϕ4gϕ2gϕ2g0+0−ηP0−η−η2E=−e12−eZ12−e∂k(εjkZjϕ).(1.4.39)8sin2θ2sinθ2cosθ2cosθAsamatteroffact,wecanuse(1.4.37)toexpressEequivalentlyasfollows:g2ϕ4gϕ2g
E=−00−ηP−η22−η22+e12+eZ12ϕ−ϕ0+2e|∇ϕ|8sinθ2sinθ2cosθg2ϕ4g2
2=0+ϕ2−ϕ2+g2ϕ2|W|2e−η+2e−η|∇ϕ|2.(1.4.40)24cos2θ08sinθ 1.4Selfdualityintheelectroweaktheory41Therefore,contrarytothepreviouscase,itmakesgoodsensenowtoconsiderplanarselfgravitatingelectroweakstrings(parallelinthex3-direction)satisfyingthefiniteenergycondition!Eeη<+∞,(1.4.41)R2whichcanbeverifiedbyrequiringadeguatebehaviorofηatinfinity.Itisinterestingtonotethat(1.4.39)leadsalsotoageometricalpropertyabout1e−η ηoftheRiemannsurface(R2,eηδtheGausscurvatureKη=−jk).Indeed,by2meansofEinstein’sequation,KηrelatestoEbytheidentityKη=8πGE+,withthecosmologicalconstant,which(forconsistency)mustbespecifiedasπGg2ϕ4=0.sin2θSo,wecaninterpretthefiniteenergycondition(1.4.41)as(almost)equivalenttoafinitetotalGausscurvatureproperty.Rigorousanalyticalresultsconcerningtheexistenceofselfgravitatingelectroweakstrings,includingsolutionsto(1.4.37),(1.4.38),and(1.4.41)(withfinitetotalGausscurvature)havebeenestablishedrecentlyin[ChT2],andwillbediscussedinChapter7. 2EllipticProblemsintheStudyofSelfdualVortexConfigurations2.1EllipticformulationoftheselfdualvortexproblemsTheexamplesofselfdualproblemsdiscussedinthepreviouschapterallshareacom-monequation(see(2.1.1)below),whichcanbeviewedasagauge-invariantversionoftheCauchy–Riemannequation.FollowinganapproachintroducedbyTaubes(cf.[JT]),weseenexthowtousesuchapropertyinordertoeliminatethegaugeinvariancefromtheselfdualequationsandformulatethemintermsofellipticproblemsoftheLiouville-type,whoseanalysiswillbethemainobjectiveofourstudy.Tobemoreprecise,letφ∈Cbeasmoothcomplex-valuedfunctiondefinedinR2(tobeidentifiedwithCwhennecessary),andA=(Aj)j=1,2;beasmoothrealvectorfieldsuchthat,D2.(2.1.1)±φ:=(∂1±i∂2)φ−i(A1±iA2)φ=0inRSince(2.1.1)isinvariantundertheabelian-gaugetransformation,iωφ,Aφ−→ej−→Aj+∂jω,j=1,2(2.1.2)(foranygivensmoothrealfunctionωdefinedoverR2),wemaysupposeAtobespecifiedaccordingtotheCoulumbgauge,whereAdefinesadivergence-freefield,namely,∂1A1+∂2A2=0.Thus,ifweletηbearealfunctionsuchthat,∇η=±(−A2,A1),(2.1.3)thenψ=e−ηφsatisfies(∂1±i∂2)ψ=0.Soaccordingtowhetherwechoosethe+or−sign,wefindthatψorψisholomorphic.Therefore,ifφisanon-trivialsolutionof(2.1.1),thenφadmitsonlyanisolatednumberofzeroeswithintegralmultiplicity.Fromthepointofviewofthevortexprob-lem,suchzeroesplaytheroleofdefectsandareresponsiblefortheoccurenceofnon-trivialphenomena. 442EllipticProblemsintheStudyofSelfdualVortexConfigurationsAssumingthat{z1,...,zN}arethezeroesofφ,repeatedaccordingtotheirmulti-plicity,andsetting⎛⎞−1$Nh(z)=e−η(z)φ(z)⎝z−zj⎠j=1or⎛⎞−1$Nh(z)=e−η(z)φ(z)⎝z−zj⎠,j=1weseethathdefinesanevervanishingholomorphicfunction.Furthermore,bythechangeofgaugeφ−→|h|h−1φ,wefindthatφtakestheformNφ(z)=|φ(z)|e±ij=1Arg(z−zj)(2.1.4)with$N|φ(z)|=eη|h(z)|z−z.(2.1.5)jj=1Inthisgauge,equation(2.1.1)readsas⎛⎞N∂2log|φ|+∂1⎝Argz−zj⎠=±A1j=1⎛⎞(2.1.6)N∂1log|φ|−∂2⎝Argz−zj⎠=∓A2,j=1andgivesawell-definedsmoothexpressionforthecomponentsA1andA2intermsofthe(gauge-invariantquantity)|φ|.Furthermore,from(2.1.5)and(2.1.3)andthefactthatlog|h|2isaharmonicfunction,wededucethat,N2=−2 η−2log|h|2−4πδ−log|φ|zjj=1N=±2F12−4πδz.jj=1HereδpdenotestheDiracmeasurewithapoleatp∈R2.Atthispoint,wecanusetheremainingselfdualequationstodeterminelog|φ|2(andhence|φ|)asasolutionofanellipticproblem.Inthisway,wehaveeliminatedcompletelythefastidiousgaugeinvariancefromtheselfdualequationsandreducedtheirstudytothesolvabilityofsomeellipticprob-lemsinvolvingonlygauge-invariantquantities. 2.1Ellipticformulationoftheselfdualvortexproblems45Tobemoreprecise,letusstarttodiscusstheabeliancase,whereφdefinestheHiggs(matter)field,andsetu=log|φ|2.(2.1.7)Foragivensetofpoints{z1,...,zN},repeatedaccordingtotheirmultiplicity,wecantakeadvantageof(2.1.4),(2.1.6),and(2.1.8),todefine1Nφ(z)=e2u(z)±ij=1Arg(z−zj),(2.1.8)and⎛⎛⎞⎞N1A1=±⎝∂2u+∂1⎝Arg(z−zj)⎠⎠,2j=1⎛⎛⎞⎞(2.1.9)N1A2=∓⎝∂1u−∂2⎝Arg(z−zj)⎠⎠.2j=1Basedontheargumentsabove,(φ,A)withA=−iAαdxαdefinesavortexsolutionfortheabelianMaxwell–Higgsselfdualequations(1.2.25),(1.2.26),and(1.2.27),providedwetakeφin(2.1.8)andA1andA2in(2.1.9)suchthatusatisfiesNu−4πδ−u=1−ez,(2.1.10)jj=1andlet,A0=0.(2.1.11)Similarly,weobtainavortexsolutionforthepureChern–Simons–Higgsselfdualequations(1.2.45),providedwenowtakeuin(2.1.8)and(2.1.9)tosatisfyN4u2u−u=e(ν−e)−4πδz,(2.1.12)k2jj=1andrecalling(1.2.46),let12uA0=±(ν−e).(2.1.13)kSimilarly,toderiveavortexsolutionfortheMaxwell–Chern–Simons–Higgsselfdualequations(1.2.63),weneedtotakeuandtheneutralscalarfieldNtosatisfy⎧⎪⎪N⎪⎪−u=2q2kN−eu−4πδz,⎨jj=1(2.1.14)⎪⎪⎪⎪1ν2−N=2eu−N+kq2(eu−kN),⎩2qk 462EllipticProblemsintheStudyofSelfdualVortexConfigurationsandrecalling(1.2.61),letν2A0=±−N.(2.1.15)kNotethatforavortexsolutionconstructedinthismanner,theHiggsfieldvanishesex-actlyat{z1,···,zN}accordingtothegivenmultiplicity.Furthermore,forthesolutionuofanyoftheellipticequationsintroducedaboveholdsthedecompositionNu(z)=logz−z2+smoothfunction,(2.1.16)jj=1andsowemaycheckthatthepotentialfieldcomponentsA1andA2intheright-handsideof(2.1.9)extendsmoothlyat{z1,...,zN},thezeroessetofφ.Analogously,forthenon-abelianpureChern–Simonsmodelin(1.3.99),(1.3.100),and(1.3.101),wemayderiveaselfdualvortex(A,φ)undertheansatzA=AαdxαwithAα=−iAaαHa,Aaα∈Randφ=φaEaφa∈C,(2.1.17)whereHaandEa,fora=1,...,rarerespectivelytheCartanalgebrageneratorsandthesimplerootoperatorsforthe(semisimple)gaugegroupGwithrankr.ForthegivenintegersNa∈Nandthesetofpoints{za,...,za}(repeatedaccordingtotheir1Namultiplicity)set
1uNaz−zaφa=e2a±ij=1Argj,(2.1.18)a=1,...,r.LetthecomponentsAa,j=1,2bedefinedintermsoftheinvertiblejCartanmatrix,K=(Kab)in(1.3.79),throughtheidentity:⎛⎞1Na
KbaAb=±⎝∂2ua+∂1Argz−za⎠,12jj=1⎛⎞1Na
(2.1.19)KbaAb=∓⎝∂1ua−∂2Argz−za⎠,22jj=1a=1,...,r.Recalling(1.3.124),wecanalsoobtainthecomponentAabymeansoftherelation:0
b12ubKbaA0=±ν−Kbae,a=1,...,r.(2.1.20)kInthisway,wedetermineavortexsolutionforthenon-abelianChern–Simonsself-dualequations(1.3.115)(reducedto(1.3.123)viatheansatz(2.1.17))provided(ua)a=1,...,rdefinesasolutionfortheellipticsystem: 2.1Ellipticformulationoftheselfdualvortexproblems474
N−ua=ν2Kbaeub−KbaeubKcbeuc−4πδa.(2.1.21)k2zjj=1IfG=SU(n+1),thentherankr=nandtheCartanmatrixK=(Kab)isexplicitlygivenby(1.3.79).Hence,inthiscase,wecaninterpretthesystem(2.1.21)asthenat-uralextensionofthesingleChern–Simonsequation(2.1.12)toaTodalatticesystemcoupledbytheSU(n+1)Cartanmatrix(1.3.79).Concerningelectroweakvortices,obtainedundertheansatz(1.4.26)–(1.4.29)assolutionsoftheselfdualequations(1.4.34),wecanuseasimilarapproach,andforassignedpoints{z1,...,zN}(repeatedaccordingtotheirmultiplicity)weset,
1u+NArgz−zaW=e2j=1j.(2.1.22)Weverifythefirstequationin(1.4.34),namely,
DA3+iA3=01W+iD2W=(∂1+i∂2)W−ig12providedusatisfies
N3−∂3−u=2g∂1A2A=2g(P12sinθ+Z12cosθ)−4πδz21jj=1whererecalling(1.4.18)and(1.4.19),wehavetaken⎛⎞N31⎝1⎠,A1=∂2u+∂1Argz−zjg2j=1⎛⎞(2.1.23)N31⎝1⎠,A2=−∂1u−∂2Argz−zjg2j=1definedsmoothlythroughthepointszj,j=1,...,N.Furthermore,takingintoaccountthelastequationin(1.4.34)and(1.4.35),welet,2vcosθϕ=eandZj=−εjk∂kv.(2.1.24)gThus,using(2.1.22),(2.1.23),and(2.1.24),weobtainanelectroweakvortexcon-figurationsolutionoftheselfdualequation(1.4.34),withWvanishingexactlyat{z1,...,zN},providedthepair(u,v)satisfies:⎧⎪⎪N⎪⎨−u=4g2eu+g2ev−4πδz,jj=1(2.1.25)⎪⎪2g2ϕ2⎪⎩2ugv0− v=−2ge−e+.2cos2θ2cos2θObservethatthefieldsBjandPjmayberecoveredfromA3in(2.1.23)andfromZjjin(2.1.24)via(1.4.18)and(1.4.19),forj=1,2. 482EllipticProblemsintheStudyofSelfdualVortexConfigurationsAnalogously,weproceedtoderiveselfgravitatingelectroweakstrings,whichin-cludetheconformalfactorηintheselfdualequationsgivenin(1.4.37)and(1.4.38).Asabove,weseethatthiscasereducestosolvingthefollowingellipticsystem:⎧⎪⎪N⎪⎪−u=4g2eu+g2ev+η−4πδz,⎪⎪j⎪⎪j=1⎨2
− v=gϕ2−eveη−2g2eu,(2.1.26)⎪⎪2cos2θ0⎪⎪2⎪⎪⎪⎪
ηg2ev−ϕ2ϕ4−=0+0eη+2g2eu+v+|∇v|2ev.⎩8πG2cos2θ2sinθThus,fortheselfdualmodelsconsideredabove,wecanobtainvortexconfigura-tions,withaprescribedsetofzeroesforthecomplexfield,bysolvingcertainellipticproblems.Atthispoint,itisimportanttobemorespecificabouttheboundaryconditions.Clearlyourchoiceofboundaryconditionsmustbesignificativefromthepointofviewofthephysicalapplications,asforexampleinidentifyingrelevantquantitiessuchasthetotalenergy,themagneticflux,andtheelectriccharge.Inthisrespect,thefirstsetofboundaryconditionsweshallconsider,concernswithplanarselfdualvortices,wheretheselfdualequations(orequivalently,thecorre-spondingellipticproblems)areexaminedinthewholeplaneR2undersuitabledecayassumptionsatinfinity,thatguaranteefinitetotalenergy.Thesecondsetofboundaryconditionsweconsiderpertainstoperiodicity.Notethatperiodicpatternsofvortexconfigurationshavebeenobservedexperimentallytoforminsuperconductivity,andthattheyareknownastheAbrikosov’s“mixedstates,”inviewofAbrikosov’spioneeringworkaboutsuperconductors(cf.[Ab])wheresuchconfigurationswerefirstpredicted.Therefore,forthe(gauge-invariant)fieldswerequirethe’tHooftperiodicbound-aryconditions(cf.[’tH2])overthecelldomain1+s22=z=s1a2a∈R,00andδzrepresentingtheDiracmeasurewithapoleatz0∈D,thenitis0possibletoobtainananalogouscharacterizationtothesolutionsof(2.2.4)bymeansofaLiouville-typeformulavalidforthepunctureddisk,asobtainedbyChou–Wanin[CW].Forthiscaseaccordingto(2.2.2),for(1)mustvanishinDexactlyatthepointfz0withmultiplicityα.Soforα/∈N,wemustallowftobemultivaluedinD.Moreprecisely,onthebasisof[CW],Bartolucci–Tarantellohaveshownin[BT1]thateverysolutionof(2.2.4)takestheformof(2.2.3)withanunivalentfunctionfinD{z0}givenbyoneofthefollowingexpressions:(i)f(z)=(z−z0)α+1ψ(z)+a,witha∈Canda=0ifα/∈N;(2.2.5)ψ(z)(ii)f(z)=;(2.2.6)(z−z0)α+1(iii)limitedtothecaseα=m−1withm∈N:f(z)=e(z−z0)α+1ψ(z),(2.2.7)2whereψisaholomorphicfunctioninDsatisfyingψ(z0)=0.See[BT1]fordetails.Basedontheprecedingobservations,theequationexhibitdifferentclassesofso-lutions.Forinstance,inthecaseoftheplanarsingularLiouvilleequation(wetakez0=0forsimplicity):u−4παδ2−u=ez=0inR(2.2.8)bychoosingf(z)=µzα+1,µ=0,andsettingλ=µ2weobtain8λ(α+1)2|z|2αu(z)=log2,∀λ>0,(2.2.9)1+λ|z|2(α+1)whichsolves(2.2.8).Or,letf(z)=µzα+1eg(z)withg(z)=(α+1)'zeξ−1dξandµ=0;thenfor0ξλ=µ2,wealsoseethat,8λ(α+1)2|z|2α|eg(z)+z|2u(z)=log2,∀λ>0(2.2.10)1+λ|z|2(α+1)|eg(z)|2solves(2.2.8).Thepresenceofthefreeparameterλin(2.2.9)and(2.2.10)canbejustifiedbythescaleinvarianceof(2.2.8)underthetransformation:u(z)−→uµ(z):=u(µz)+2logµ,∀µ>0. 542EllipticProblemsintheStudyofSelfdualVortexConfigurationsSuchaninvariancewillbeattheoriginof“concentration”phenomenaforthesolu-tionsequencesofLiouville-typeequations,aswillbediscussedingreatdetailinChapter5.Asalreadyshownby(2.2.5),whenα=N∈Z+,wegainanadditionalfree(complex)parameter.Infact,wecangeneralize(2.2.9)inthiscasebytakingf(z)=µ(zN+1+ξ),∀ξ∈C,andthensetλ=µ2toobtainthesolution:8λ(N+1)2|z|2Nu(z)=log2,∀λ>0,ξ∈C.(2.2.11)1+λ|zN+1+ξ|2Noticethatforα=N=0,thefreeparameterξ∈Cisdueonlytothetranslationinvarianceof(2.2.8).Ifinsteadα=N=0,thentheadditionalparameterξ∈Cwouldhavestrongconsequencesonthenatureofthesolutions.Infact,thisparameterisresponsibleforsymmetry-breakingphenomena;aswecanobtainsolutionsof(2.2.8)whicharenolongerradiallysymmetricaboutanypoint.Thepresenceofnon-radialsolutionsforthisclassofequationswasfirstnoticedbyChanillo–Kiesslingin[CK1].Itisclearthatotherchoicesoffin(2.2.3)wouldyieldtoyetotherclassesofsolutionsfor(2.2.8);seee.g.,Section5.5.5inChapter5.However,theexamplesabovedistinguishbetweentwoimportantclassesofsolutions—namely,thosesatisfyingeu∈L1(R2)asgivenby(2.2.9)and(2.2.11),andthosewitheu∈/L1(R2)asgivenin(2.2.10).Infact,asobservedin[CW]forα=0,andsubsequentlyin[PT]forα>0,thefiniteenergycondition,!eu<+∞,(2.2.12)R2onlyallowsafunctionfofthe“power-type”in(2.2.3).Moreprecisely,thefollowingclassificationresultholds:Theorem2.2.1([CL1],[CW],[PT])Forα≥0,everysolutionuof(2.2.8)satisfying(2.2.12)takestheform8λ(α+1)2|z|2αu(z)=log2,(2.2.13)1+λ|zα+1+ξ|2withλ>0,ξ∈C,andξ=0forα/∈Z+.Inparticular,!eu=8π(1+α).(2.2.14)R2Weemphasizethatforα/∈Z+,wemusttakeξ=0in(2.2.13);otherwisewewouldobtainamultivaluedsolution. 2.2ThesolvabilityofLiouvilleequations55Asalreadymentioned,suchclassificationresultisaconsequenceofLiouvillefor-mula(2.2.3)togetherwith(2.2.5)–(2.2.7),andwereferto[CW]and[PT]fordetails.Letusmentionthat,thecaseα=0washandledfirstbyChen–Li[CL1],bythemethodof“movingplanes”ofAlexandroffinthesamespiritof[GNN]and[CGS].Namely,in[CL1]theauthorsshowthatforα=0,allsolutionsof(2.2.8)and(2.2.12)areradi-allysymmetricaboutapoint,andconsequentlyarriveat(2.2.13)(withα=0)afteranO.D.E.analysis.Clearly,suchanapproachcannotworkingeneralforthecaseα>0,sinceradialsymmetrymaybebrokenbythesolutionsof(2.2.8)whenα∈N.Nevertheless,themereidentity(2.2.14)canbederivedfromageneralsymmetryresultobtainedbyChen–Li[CL2]withintheframeworkofthe“movingplane”analy-sis,appliedtov(theregularpartofu)anddefinedasfollows:v(z)=u(z)−2αlog|z|.(2.2.15)Noticethatvsatisfies:− v=|z|2αevinR2,'(2.2.16)2|z|2αev<+∞.RInfact,formanypurposesitismoreconvenienttoanalyzeproblem(2.2.16)ratherthan(2.2.8)and(2.2.12).Intermsof(2.2.15),Theorem2.2.1readsasfollows:Corollary2.2.2Letα≥0,theneverysolutionof(2.2.16)takestheform:λ1v(z)=log2,γα=2,(2.2.17)1+λγα|zα+1+ξ|28(1+α)withλ>0,ξ∈C,andξ=0forα/∈Z+.Furthermore,!|z|2αev=8π(1+α).(2.2.18)R2Observethat,thescaleinvarianceof(2.2.16)isexpressednowaccordingtothetrans-formation:v(z)−→vµ(z):=v(µz)+2(α+1)logµ,∀µ>0.Wepointoutasymmetrypropertyforthesolutionsof(2.2.16),whichisnotatallobviousfromexpression(2.2.17)whenα∈Nandξ∈C{0}.Tothispurpose,foragivensolutionvof(2.2.16),weletv∞=limv(z)+4(α+1)log|z|.(2.2.19)|z|→+∞By(2.2.17),thelimit(2.2.19)existsandisfinite.Proposition2.2.3Letα≥0andvbeasolutionof(2.2.16).Setv(0)−v∞τ=exp2(α+1) 562EllipticProblemsintheStudyofSelfdualVortexConfigurationswithv∞in(2.2.18).Thenz12v(z)=v+2(α+1)log,inR.τ|z|2τ|z|2Furthermore,setting|z|=r,wehave11r−√(r∂rv+2(α+1))<0,ifr=√ττ(2.2.20)1r∂rv+2(α+1)=0,ifr=√.τProperties(2.2.19)and(2.2.20)wereestablishedinamoregeneralcontextin[PT,Theorem2.5],towhichwereferthereaderfordetails.2.3VariationalframeworkAsweshallsee,manyoftheellipticproblemsweshallanalyzebelowadmitausefulvariationalformulation,sothatthesearchfortheirsolutionscanbereducedtofindingthecriticalpointsofaFrechetdifferentiablefunctional,I:E−→RwithEasuitableBanachspace.Recallthatv∈EisacriticalpointforIinEwithacriticalvaluec=I(v),ifwehave||I(v)||E∗=0.Asusual,withE∗wedenotethedualspaceofE,sothatforacriticalpointvwehavemax(I(v),ϕ)=0.(2.3.1)ϕ∈E∗Typically,theconstructionofaso-calledPalais–Smale(PS)-sequence,namelyase-quence{vn}⊂Esuchthat(v||In)||E∗→0,andI(vn)→c(2.3.2)representsthefirststepinobtainingacriticalpointwithacriticalvaluec.Forinstance,ifIisboundedfrombelow,then(2.3.2)mayberealizedbyaminimizingsequenceandc=infEI.See,forexample[Gh].Or,whenIadmitsa“mountain-pass”structure(cf.[AR]),inthesensethat∃⊂Eaclosedsetwhichseparatestwopointsv0andv1∈Eand(2.3.3)infI>max{I(v0),I(v1)}.Then(2.3.2)mayberealizedviaamin-maxprocedure,byconsideringthesetofallcontinuouspathsjoiningv0andv1,(namely,P:={γ:[0,1]−→Econtinuouswithγ(0)=v0andγ(1)=v1}) 2.4Moser–Trudingertypeinequalities57andbysettingc=infmaxI(γ(t))>max{I(v0),I(v1)}.(2.3.4)γ∈Pt∈[0,1]Inthiscase,thecorresponding(PS)-sequenceisobtainedbyusingtheflowassociatedtothepseudogradientvectorfieldrelativetoI.Moreprecisely,Theorem2.3.4LetI∈C1(E),andassumethatthereexistc>0andδ>0suchthat(v)||ifv∈E:|I(v)−c|<δthen⇒||IE∗≥δ.(2.3.5)Thenforanyε>0sufficientlysmallandsuitableε∈(0,ε),wefindadeformationmapη:E×[0,1]−→E(2.3.6)satisfying:(i)η(v,0)=vandI(η(v,t))isdecreasingint∈[0,1];(ii)ifv∈E:|I(v)−c|≥ε,thenη(v,t)=v∀t∈[0,1];(2.3.7)(iii)ifv∈E:I(v)0andγ>0suchthat,foreveryv∈H1(R2)thereholdsγ||v||2||ev−1||H1(R2).(2.4.3)L2(R2)≤CeMoreover,foreveryp≥2themap

H1R2−→LpR2(2.4.4)v−→ev−1iscontinuous.Proof.Toestablish(2.4.3),weusetheTaylorexpansionandwrite+∞2k−1−1v2ke−1=2v.k!k=2So,bymeansof(2.4.2)wecanestimate:+∞2k−1−1ev−12≤2||v||kL2(R2)k!Lk(R2)k=2+∞2k−1−1k−2k−22≤2||v||k(2.4.5)k!2H1(R2)k=2+∞kk21≤2(2π)k||v||k.2k!H1(R2)k=2 2.4Moser–Trudingertypeinequalities59SincebySterling’sformula,wemayfindconstantsc2>c1>0suchthatk≤k!≤(ck(c1k)2k),from(2.4.5),wereadilyarriveat(2.4.3).Thus,ev−1∈L2(R2)and(2.4.4)followsforp=2.Ontheotherhand,foreveryn∈N,wehavennn

vn!kvn−kn!n−kkv22e−1=e(−1)=(−1)e−1∈LR;k!(n−k)!k!(n−k)!k=0k=0thatis,v2n2vγn||∇v||2e−1∈L(R)ande−1L2n(R2)≤CneL2(R2)foreveryn∈N,withCn>0andγnsuitableconstants(dependingonn).Therefore,byinterpolationweconcludethat(2.4.4)holdsforanyp≥2.TheideaofusingSobolev’sinequalitieswithexplicitappropriateconstantstoestimatethenormoftheexponentialwasadoptedfirstbyTrudingerin[Tr]intheframeworkoftheSobolevspaceH1()with⊂R2abounded(regular)domain0(seealso[Sal]).Subsequently,Trudinger’sresultwasre-derivedbyMoserinamoregeneralformasfollows:Proposition2.4.8([Tr],[Mo])Forn≥2thereexistsaconstantCn>0(dependingonnonly)suchthatsetting1n−1αn=nω(2.4.6)n−1withωnthevolumeoftheunitsphereSninRn+1,wehave(!)nαn|v|n−11,nsupe,v∈W0()and||∇v||Ln()=1≤Cn||,(2.4.7)where||istheLebesguemeasureof⊂Rn.Furthermore,αnin(2.4.6)isthebestpossibleconstantforwhichthesupremumintheleft-handsideof(2.4.7)isfinite.Here,weareinterestedinconsidering(2.4.7)forthecasen=2,whereitstatesthat,2!v4π1||∇v||∀v∈H0():eL2()≤C||(2.4.8)withauniversalconstantC>0andindependentof⊂R2.From(2.4.8)weeasilyderivetheestimate!1|∇v|2−ev≤Ce16πL2(),∀v∈H1();(2.4.9)0 602EllipticProblemsintheStudyofSelfdualVortexConfigurationswhereweusestandardnotationtodenotetheaveragevalueoffas!!1−f:=f.(2.4.10)||Theinequality(2.4.9)furnishesa“sharp”versionof(2.4.3)forboundeddomains.Infact,thevalue1isthebestpossibleforinequality(2.4.9)tohold.Thiscanbe16πcheckeddirectlywiththehelpofthe“concentrating”solutionsoftheregularLiouvilleequation(i.e.,(2.2.8)withα=0).Indeed,byassumingforsimplicity(andwithoutlossofgenerality)thatB1(0)⊂andbysetting
2log1+λif|z|≤1vλ(z)=1+λ|z|2∈H1(),(2.4.11)00if|z|≥1wecancheckwithoutgreatdifficultythat,asλ→+∞,||∇v2=16πlog(1+λ)+o(1)(2.4.12)λ||2L()and!log−evλ=log(1+λ)+O(1).(2.4.13)Therefore,asλ→+∞,vλfurnishesan“optimal”familyfor(2.4.9)andshowsthat(2.4.9)doesfailifwerepace1withasmallervalue.16πTheexampleabovealsoindicatesthatitmaynotbealwayspossibletoobtainanextremalfunctionfor(2.4.9).Non-existenceofanextremalfunctioncanbecheckedwhenisaball,whileexistencemayoccurforannulusorrectangulardomains(see[Ban],[Su1],[CLMP1],and[CLMP2]).Onthecontrary,itissurprisingtoseethattheexistenceofaminimizercanalwaysbeguaranteedfortheoriginalinequality(2.4.8),asithasbeenshowntoin[CaCh],[Fl2],[Ln],and[St4].Itisofinteresttoustoconsiderpossibleversionsof(2.4.8)and(2.4.9)overcom-pactRiemannsurfaces(M,g).WeconsiderthecasewhereMhasnoboundary(i.e.,∂M=∅)anddenoteby∇ganddσg,respectively,thecovariantderivativeandthevolumeelementinducedbytheRiemannmetricgonM.Theinequality(2.4.8)continuestoholdwithintheframeworkoftheSobolevspaceH1(M);thatis(!)4πu21supedσg,u∈H(M)and||u||H1(M)=1<+∞(2.4.14)M(see[Au],[Ad],[yLi1],and[yLi2]forgeneralizations).Andasabove,ityieldstotheinequality!
1||∇22u16πgu||L2(M)+||u||L2(M)edσg≤Ce.M 2.4Moser–Trudingertypeinequalities61Inparticular,weseethatforeveryp≥1,themapH1(M)−→Lp()u(2.4.15)u−→eiscontinuous.Ontheotherhand,onthesubspace(!)1(M):wdσE=w∈Hg=0,(2.4.16)Mthenorm||w||H1(M)and||∇gw||L2(M)areequivalent,anditispossibletoshowthattheexactsameinequalityasin(2.4.7)holds(see[Fo]).Namely,Proposition2.4.9([Fo])LetMbeacompactRiemannsurface.ThereexistsaconstantC>0suchthat!1||∇2w16πgw||L2(M)−edσg≤Ce,∀w∈E.(2.4.17)MInterestinglyenough,forthestandard2-sphereM=S2,theinequality(2.4.17)canbederivedasalimitingcase(forp→+∞)ofthewell-known(sharp)Sobolevinequality2p−221212||u||Lp(S2)≤1−2||∇u||L2(S2)+1−2||u||L2(S2),u∈H(S),(2.4.18)pp2ωω22validforp≥2andω2=|S2|=4π(seee.g.,[Be]foraversionof(2.4.18)inthedimensionn≥3).Indeed,byapplying(2.4.18)withu=1+1wandw∈E,wefindp⎛⎞p!2!2!1p⎝p−22p12⎠(1+w)dσ≤|∇w|dσ+ω+wdσ1−221−2S2pp2S22pS22ωppω22!!p2p−2212=ω21+2|∇w|dσ+2wdσ.2ω2pS2ω2pS2Hence,bypassingtothelimitasp→+∞,wederivethedesiredMoser–TrudingerinequalityforfunctionsdefinedoverS2:!!11||∇w||2ewdσ≤e16πL2(S2),∀w∈H1(S2):wdσ=0(2.4.19)2|S|S2S2(see[Mo],[On],[Au]andreferencestherein).TheproofaboveisduetoBecknerin[Be],whichcontainsvariousothergeneralizations.Thederivationof(2.4.17)forgeneralRiemannsurfaceswasobtainedbyFontanain[Fo],alongthelinesof[Ad].Seealso[DJLW1],[NT2],[ChCL],[Che],and[ChY3]forrelatedresults. 622EllipticProblemsintheStudyofSelfdualVortexConfigurationsWemayrelate(2.4.19)toaninequalityoverR2(tobecomparedwith(2.4.3))viathesterographicprojection.Tothispurpose,considerS2embeddedinR3,asgivenbythesetofallx∈R3:||x||=1.Definethe(inverse)stereographicprojectionwithrespecttothesouthpole(0,0,−1)asπ:R2−→S2{(0,0,−1)}z=(x,y)−→(ρz,t)21−|z|2withρ=1+|z|2andt=1+|z|2.Thus,foreveryv∈H1(S2),wemayconsiderafunctionu=v◦πdefinedoverR2andverifythat:!!2p2p|v|dσ=|u|dxdy,∀p≥1;2S2R21+|z|!!2v2u(2.4.20)edσ=edxdy;2S2R21+|z|!!|∇v|2dσ=|∇u|2dxdy.S2R2Therefore,ifweset(

)22u22E=u:|∇u|∈LRand∈LR,(2.4.21)1+|z|2thenv∈H1(S2)ifandonlyifu=v◦π∈E,!!uvdσ=0ifandonlyif2dxdy=0,S2R21+|z|2and(2.4.19)reducesto!!1eu121u2≤exp||∇u||L2(R2)+2.(2.4.22)πR21+|z|216ππR21+|z|2Theseobservationspermitustousetheclassificationresult(2.2.17)withα=0,toconcludethefollowingresultfirstnoticedbyOnofriin[On]:Proposition2.4.10([On],[Ho])Theequalityin(2.4.19)isattainedifandonlyifw=0.Proof.Assumethatwattainstheequalityin(2.4.19).Withoutlossofgenerality,wemaysupposethatmax'S2wisattainedatthenorthpole(0,0,1).Therefore,thefunctionu=(w◦π−log1ewdσ)∈Eattainsequalityin(2.4.22)andsatisfies4πS2 2.4Moser–Trudingertypeinequalities63⎧⎪⎪8u2⎪⎨−u=2(e−1)inR1+|z|2!u⎪⎪e⎪⎩u(0)=maxu,2=π.R2R21+|z|2Consequently,setting8U(z)=log2+u(z),1+|z|2weseethatitsatisfies⎧⎨−U=eUinR2!⎩U(0)=maxU,eU=8π.R2R2Thus,wecanuseCorollary2.2.2,withα=0,toconcludethatbynecessity:8λU(z)=log,forsomeλ>0.(1+λ|z|2)2Inotherwords,1+|z|2u(z)=logλ+2log.1+λ|z|2Since|∇u|∈L2(R2)λ=1musthold,whichimpliesu=0;andsow=0asclaimed.AnalogousinequalitiesoverR2havebeenderivedbyMcOwen[McO]withotherweightfunctions.Wealsoreferto[DET]formoregeneralweightedinequalitiesofthe(2.4.22)type,andtheirinfluenceinthesymmetry-breakingphenomenaoftheHardy-Sobolevinequality(cf.[CKN]).VersionsoftheabovementionedinequalitiesareavailablealsoforthecasewhenMadmitsanon-emptyboundary∂M=∅(see[Fo]and[Au]andreferencestherein).However,inthiscase,theconstant1isnolongerappropriate.Weknowthatfor16πasmooth∂M,theconstantmustbereplacedby1,whileitsvaluebecomesmore8πinvolvedwhen∂Madmitscorners,asithasbeendiscussedin[ChY1].Formanifoldsotherthanthe2-sphere,theexistenceofaminimumfortheextremalproblem(2.4.14)canalwaysbeensured(see[yLi1]and[yLi2]).However,weencounteradelicateproblemwheninvestigatingthepossibilityofattainingequalityin(2.4.17).Ofparticularinteresttousisthecaseoftheflat2-torusM=R2
a1Z×a2Zwherea1anda2aretwolinearlyindependentvectorsofR2thatgeneratetheperiodiccelldomain={sa1+ta2,00.Stickingtoourcomplexnotations,wearegoingtoidentifythe(independent)vec-torsaj=αje1+βje2withthecomplexnumberwj=αj+iβj,j=1,2,sothatiw1,w2=Re(iw1w2)=0formulatestheconditionforlinearindependency.Inthisway,wemayexpresstheflattorusasM=C
(w1Z+w2Z).Bytheworkin[DJLW1]and[NT2],weknowthatforM=C
(aZ+ibZ)and00,wemayconsiderthefunctional!12wIµ(w)=||∇gw||2−µloghedσg,w∈E(2.5.1)2L(M)M(recallEin(2.4.16)).Byvirtueof(2.4.15),thefunctionalIµ∈C1(E),anditscriticalpointscorrespondto(weak)solutionsforthemeanfieldequationoftheLiouvill-type
'hew1−gw=µhewdσg−|M|inM'M(2.5.2)wdσg=0MwithgtheLaplace–BeltramioperatorcorrespondingtotheRiemannianmetricgonM,and|M|thesurfaceareaofM.Weshallbeinterestedinhandling(2.5.1)or(2.5.2)underthefollowingsetofassumptionsonh:!u0∈L∞(M):u1h=e0∈L(M)andu0dσg=0.(2.5.3)MNoticethatthelastconditionin(2.5.3)impliesnorealrestrictiononh,sinceproblem(2.5.2)remainsunchangedifwereplacehwithth,t>0.Equation(2.5.2)hasattractedmuchattentioninthelasttwodecadesbythecentralroleithasplayedinavarietyofproblemsarisinginconformalgeometry(seee.g.,[Au],[Ba],[ChY1],[ChY2],[ChY3],[CL],[CD],[CL3],[CK3],[H],[Ho],[K],[KW1],[KW2],[L1],[Ob],[Ni],[On],andthereferencestherein),mathematicalphysics(see 2.5AfirstencounterwithmeanfieldequationsofLiouville-type65e.g.,[CLMP1],[CLMP2],[CK1],[CK2],[CK3],[On],[Ki1],[Ki2],[Wo],andthereferencestherein)andappliedmathematics(e.g.,[Cha],[Ci],[CP],[BE],[Ge],[KS],[EN],and[Mu]).Inourcontext,problem(2.5.2)hasenteredinacrucialwayintheunderstandingoftheasymptoticbehaviorof“non-topological”Chern–Simonsvorticesandinthestudyofelectroweakmixedstates.Thesolvabilityof(2.5.2)posesaratherdelicateproblem,aswecanseealreadyonthebasisof(2.4.24).Thisleadsustodistinguishthefollowingcases:Case1:µ∈(0,8π),thenthefunctionalIµiscoerciveandboundedfrombelowinE.Sinceitisalsoweaklylowersemicontinuous,itattainsitsinfimumatasolutionof(2.5.2).Hence,problem(2.5.2)isalwayssolvableinthiscase.Actually,fortheflat2-torus,where(2.5.2)reducestoaperiodicboundaryvalueproblem,itispossibletousetheWeierstrassP-functionintotheLiouvilleformula(2.2.3)toexhibitanexplicitsolutionwhenµ=4π.Fordetailssee[Ol].Inthiscase,itisimportanttounderstandunderwhichcircumstanceswecanclaimuniquenessofthesolution(oroftheminimizer).Whenhisaconstant,thisamountstoaskingifproblem(2.5.2),withµ∈(0,8π),admitsonlythetrivialsolutionw=0.Weknowtheanswertobeaffirmativeforthe2-sphere(cf.[On],[Ho],[CK1],and[Li1]);whereasmultiplicitydoesoccurfortheflat2-torus,M=C
(aZ+ibZ)and08π,thenthefunctionalIµisunboundedinE,andmin-maxcrit-icalvaluesmustbesoughtinordertoobtainsolutionsfor(2.5.2).Following[ST],[DJLW3]and[BT2],weshallgiveanexampleofsuchmin-maxconstructioninSection6.2ofChapter6.Noticethatinthiscasewefacethedifficultyofcheckingthatthe(PS)-conditionholdsforIµ.Tothisend,weuseStruwe’smonotonicitytrick(cf.[St1],[St2],[St3],and[Je]),asweseethatIµisdecreasingwithrespecttotheparameterµ.Indeed,recallingJensen’sinequality,!!ifF:R−→RisconvexthenF−udσg≤−F(u)dσg,(2.5.8)MMandusingitwithF(t)=et,wefindthat!!−eu0+wdσg≥1,∀w∈Eandu0dσg=0.(2.5.9)MMThus,inviewof(2.5.3),forµ1≤µ2,wehaveIµ(w)≥Iµ(w),∀w∈E.12Weobtainanexistenceresultfor(2.5.2),whenMadmitsgenusg>0,µ∈(8π,16π)and(2.5.3)holds.Inparticularweareallowedtotakeu0asin(2.5.5).See[ST],[DJLW3],[BT2],andSection6.2inChapter6foradetailedproof.Wementionthat,whenthevalueoftheparameterµ≥16πand(beside(2.5.3)),weassumethath>0inM,thenexistenceresultsfor(2.5.2)canbededucedbythedegreeformulaobtainedbyChen–Linin[CL1]and[CL2].Moreprecisely,ifh∈C0,1(M)isstrictlypositiveonM,thenLiin[L2]showedthat,foreveryµ∈R+8πN,theLeray-SchauderdegreedµatzerooftheFredholmmapId+µTh,withThin 682EllipticProblemsintheStudyofSelfdualVortexConfigurations(2.5.6),iswell-defined.Moreover,forµ∈(8π(n−1),8πn)thedegreedependsonlyontheintegern∈N,andonthetopologicalpropertiesofM.Subsequently,Chen–Linin[ChL1]and[ChL2]wereabletocompleteLi’sanalysisandarrivedatthefollowingformula:1,ifµ∈(0,8π)dµ=(−χ(M)+1)...(−χ(M)+n−1)(2.5.10),ifµ∈(8π(n−1),8πn),n∈N{1}(n−1)!whereχ(M)=2(1−g)istheEulercharacteristicofMwithgenusg.NoticethatforanymanifoldMwithpositivegenus,inparticulartheflat2-torus,weseethatdµ>0,∀µ∈R+8πN,andsowecanensuretheexistenceofasolutionfor(2.5.2)inthiscase.Morepreciselyin[ChL2]itisshownthatfortheflat2-toruswehavedµ=1,∀µ∈(0,+∞).However,thisdoesnotimplythatthecorrespondingsolutionsareuniformlybounded,see[LiL],[LiW],and[Lu].ForthestandardsphereM=S2,wehavedµ=−1forµ∈(8π,16π),andsoexistenceisguaranteedinthiscase;whiledµ=0foranylargervalueµ/∈8πN.Thisleavesthequestionofexistenceasachallenging,openprobleminthiscase.See[Dj]forsomecontributioninthisdirection.Againwestressthattheresultsabovedonotapplywhenh=eu0andu0isgivenby(2.5.5).Indeed,inthiscase,Nu0(z)=4πG(z,zj)j=1whereG(z,p)definestheGreen’sfunctionofginH1(M),satisfying:1gG(·,p)=δp−inM,'|M|(2.5.11)G(·,p)dσg=0.MNotethatG(z,p)=G(p,z)andasiswell-known(cf.[Au]),1G(z,p)=log(dg(z,p))+γ(z,p),2πwheredg(·,·)denotesthedistancefunctiononM,andγ(theregularpartofG)isasuitablesmoothfunctiondefinedonM×M.Consequently,$Nu0(z)=d2(z,zh(z)=egj)V(z),(2.5.12)j=1withsuitable00inM,formallywecantakeN=0intheaboveformulaandreduceto(2.5.10).However,whenhvanishesasin(2.5.12)andµ∈(16π,+∞)8πN,thenageneralformulafordµisnotyetavailable.Weexpectthattheknowledgeoftheexpressionfordµwouldcarryrelevantinfor-mationabouttheN-vortexproblem.ThisweseealreadywhenMistheflat2-torus,wherewecanusetheaboveformulatoseethatdµadmitsajumpwhencrossingthevalueµ=8π,fromdµ=1(forµ<8π)todµ=N+1(for8π<µ<16π).Onthecontrary,whenN=0thendµ=1foranyµ∈(0,16π){8π}. 702EllipticProblemsintheStudyofSelfdualVortexConfigurationsOthercontributionsrelatedtoellipticproblemsinvolvingexponentialnonlineari-tiescanbefoundforinstancein[Ban],[BP],[Ch2],[DeKM],[DDeM],[Es],[EGP],[Ge],[MW],[M],[NS],[Ni],[P],[Sp],[Su1],[Su2],[WW1],[WW2],[Wes],[Wo],and[Y8].TheanalysisofPS-sequencesforIµcanbefoundin[OS1].Extensionsoftheresultsdiscussedabovetothecaseofsystemsareofmuchinter-estinapplications(seee.g.,[CSW],[SW1],[SW2]).Moreprecisely,foranassignedsymmetricinvertiblepositivedefiniten×nmatrixA=(aij)i,j=1,...,nweareinterestedinanalyzingthefunctional1!nn!Ii,jwjµ(w1,...,wn)=a∇wi∇wjdσg−µjlog−hjedσg,2MMi,j=1j=1(2.5.16)whereA−1=(aij)i,j=1,...,ndenotestheinverseofthegivenmatrixA,µ=(µ1,...,µn)∈(R+)nisan-pleofassignedpositivenumbers,hj∈L∞(M)isagivennon-negativefunction,andwj∈E,foreveryj=1,...,n.Weshallnaturallyarriveatconsideringafunctionalofthetype(2.5.16)inourstudyofnon-abelianSU(n+1)-vorticeswithmatrixA=KasgivenbytheCartanmatrixin(1.3.79)relativetothegaugegroupSU(n+1).Forlaterpurposes,recallthattheCartanmatrixK=(Kij)i,j=1,...,nisidentifiedbythecondition:Kii+1iij=2δj−δj−δj+1.(2.5.17)Theproblemtodeterminesharpconditionsonthen-pleµ=(µ1,...,µn),suchthatIµisboundedfrombelowinEnwasfirsttreatedbyChipot–Shafrir–Wolanskyin[CSW],whoactuallyconsideredtheDirichletanalogofthefunctional(2.5.16)asdescribedin(2.5.19)below.In[CSW],theauthorsassumethattheentriesofthema-trixAarenon-negative,aconditionjustifiedbytheiraimtotreatmodelsinpopulationdynamicsinabsenceofconflicts.Theyintroduce2n−1quadraticpolynomialsj(µ)definedforeverynon-emptysubsetJ⊂{1,...,n}as⎛⎞J(µ)=µk⎝8π−akjµj⎠,(2.5.18)k∈Jj∈Jandprovethat,foraregulardomain⊂R2theDirichletfunctional(tobecomparedwith(2.5.16)),1!nn!!ai,j∇wwjJµ(w1,...,wn)=i∇wj−µilog−hje−wj,2i,j=1j=1(2.5.19)isboundedfrombelowin(H1())nprovided0J(µ)>0,∀J⊆{1,...,n}.(2.5.20) 2.5AfirstencounterwithmeanfieldequationsofLiouville-type71Onthecontrary,suchaboundenesspropertyisviolatedwhentheoppositeinequalityholdsin(2.5.20),forsomeJ⊆{1,...,n}.ThisresultwasextendedbyWangin[W]forthefunctionalIµin(2.5.16).Since(2.5.20)reducesexactlytotheMoser–Trudingerconditionforn=1(see(2.4.17)),itisnaturaltoaskwhetherIµ(orJµ)remainsboundedfrombelowinEn(orin(H1())n)evenwhenweallowequalityin(2.5.20).Wethusrelaxtothecondition:0J(µ)≥0,∀J⊆{1,...,n}.(2.5.21)Inthisrespect,anaffirmativeanswerwasgivenbyWang[W]forstochasticmatricesA=(aij)i,j=1,...,nsatisfying:naij≥0,∀i,j=1,...,nandaij=1,∀i∈{1,...,n}.(2.5.22)j=1Wang’sapproachwaspursuedfurtherbyJost–Wangin[JoW1]fortheTodasystemofinteresthere,namely,whenwetakeA=K,theCartanmatrixrelativetoSU(n+1)givenin(2.5.17).Noticethat,inthiscase,(2.5.21)reducestothecondition:µj≤4π,∀j=1,...,n.(2.5.23)Wesummarizesuchresultsinthefollowing:Theorem2.5.12([W],[JW])Undertheassumption(2.5.22)orwhenA=K,thecondition(2.5.21),orrespectively(2.5.23),isnecessaryandsufficientforIµtobeboundedfrombelowinEn.Moreover,whenA=Kandequalityholdsin(2.5.23)forsomej∈{1,...,n},theexistenceofaminimizersforIµhasbeenestablishedbyJost–Lin–Wangin[JoLW],underaconditionanalogousto(2.5.4).Whereas,when(2.5.23)isviolatedandsoIµisnolongerboundedfrombelow,theexistenceofacriticalpointforIµisestab-lishedin[LN],[ChOS],[MN],and[JoLW],inthesamespiritof[ST]and[DJLW1].Wementionthatin[JoLW]onecanalsofindadegreeformulaforthecorrespondingSU(n+1)-Todasystemvalidforacertainrangeofparametersµj,j=1,...,n.ReturningtoageneralmatrixA=(aij)i,j=1,...,nwementiontheworkofShafrir–Wolansky[SW1],concerningthefunctionalIµoverM=S2.Theauthorsin[SW1]areabletoidentify,in(2.5.10),thenecessaryandsufficientconditionsuchthatIµ(withM=S2)isboundedfrombelowinEn,providedsomemildconditionholdsfortheentriesofthematrixA.Wereferto[SW1]fordetails;herewemerelymentionthattheapproachofShafrir–Wolanskyreliesona“duality”method,whichinparticularyieldstoasimpleanddirectproofoftheresultsin[W]and[JoW1].See[SW1]alsoforageneraldiscussiononthepropertiesofthefunctionalIµinrelationtothepropertiesofthematrixA. 722EllipticProblemsintheStudyofSelfdualVortexConfigurationsTheEuler–Lagrangeequationrelativetothefunctional(2.5.16)leadstothefollow-ingsystemofmeanfieldequationsoftheLiouville-typeinthevariable(w1,...,wn):⎧nw⎪⎪hjej1⎪⎨− wi=aijµj'−inM,hjewjdσg|M|!j=1M(2.5.24)⎪⎪⎪⎩widσg=0,i=1,...,n.MForproblem(2.5.24),onemaytrytoestablishpropertiessimilartothosediscussedaboveforthesingleequation(2.5.2).However,asidefromthecoercitivitycondition,(2.5.20),verylittleisknownaboutthestructureofthesolutionsetof(2.5.24).Inter-estingprogresshasbeenobtainedrecentlyforthecasewhenwetakeaij=Kijin(2.5.17);namely,AcoincideswiththeCartanmatrixKrelativetoSU(n+1).Inthiscase,(2.5.24)takesthestructureofthefollowingTodasystem:⎧n
w⎪⎪ii+1ihjej1⎪⎨− wi=2δj−δ−δj+1µj'−,jhwj|M|jedσg!j=1M(2.5.25)⎪⎪⎪⎩widσg=0,i=1,...,n.MProblem(2.5.25)isparticularlyrelevantforourpurposes,sinceitdescribesthelimitingproblemforaclassofSU(n+1)-vorticesconcerningthenon-abelianChern–Simonsmodelof(1.3.99)and(1.3.100),aswillbediscussedinChapter4.Fromananalyticalpointofview,theTodasystem(2.5.25)offerstheadvantagethatwecandetermineexplicitlyallsolutionsoftherelatedplanarproblem:−uuiinR2,i=Kije(2.5.26)i=1,...,n(Kijin(2.5.17))intermsofn-complexfunctions,bymeansofaformulathatreducestoLiouvilleformula(2.2.3)forthecasewhenn=1(see[Ko],[MOP],and[LS]).Bythisinformation,Jost–Wangin[JoW2]wereabletoobtainaclassificationresultforallsolutionsof(2.5.26)subjecttocertainintegrabilitycondition,inthesamespiritofTheorem2.2.1(orCorollary2.2.2).Thishasfurnishedthestartingpointfortheblow-upanalysisdevelopedin[JoLW]yieldingtosomecompactnessresultsanddegreeformulaeforsolutionsof(2.5.25).See[JoLW]fordetails,and[LN],[ChOS],and[MN]forpreviousresults.2.6FinalremarksandopenproblemsWeconcludethischapterwithasummaryofthemainopenproblemsconcerningthemeanfieldequationsoftheLiouville-typein(2.5.2),withparticularemphasisonthoserelatedtothestudyofChern–Simonsvortices.Westartbyconsideringthecaseforwhichtheweightfunctionh=constant. 2.6Finalremarksandopenproblems73Openproblem:LetM=T2betheflat2-torusandλ1(T2)bethefirstpositiveeigenvalueof−inH1(T2).Isittruethatproblem(2.5.2)(withM=T2,h=1)admitsonlythetrivialsolution,u=0,ifandonlyif0<µ0,
|F1−|φ|2.(3.2.3)12|+|D1φ|+|D2φ|≤c0(iii)∀>0∃C>0,0<1−|φ|0).Thisfactk2willenableustodescribetheasymptoticbehaviorofthevortexsolutionask→0. 803PlanarSelfdualChern–SimonsVorticesMorepreciselyweprove:Theorem3.2.3ForanyintegerN∈Nandanyassignedsetof(vortex)pointsZ={z1,...,zN}⊂R2(repeatedaccordingtotheirmultiplicity)andk>0,wefind(A,φ)±asmoothsolutiontotheselfdualequation(1.2.45)inR2(withthe±signchosenaccordingly)suchthatthefollowingholds.i)Propertiesofφ±:(a)φ±vanishesexactlyatthesetZ.Moreoverifnj∈Zisthemultiplicityofzj∈Zj=1,...,N,thennjφ+(z)andφ−(z)=Oz−zj,asz→zj.(3.2.5)Furthermore,(3.2.1)and(3.2.2)holdforφ±.(b)|φ±|ismonotonedecreasingwithrespecttok>0andismaximalamongallsolutionsof(1.2.45),satisfying(3.2.2)and(3.2.5).(c)Foreveryk0>0,ε∈(0,1)andδ>0,thereexistconstantsCε>0andCε,δ>0suchthatthefollowingestimatesholdforevery00suchthat∀k∈(0,kε),wehave2(1−ε)(R−|z|)|D1φ±|+|D2φ±|+|(F12)±|≤C0ek0,∀|z|≥R0,withsuitablepositiveconstantsc0,C0andR0independentofεandk.(b)Form∈Z+andδ>0,wehavec0|||D1φ±|+|D2φ±|+|(F12)±|||Cm(δ)≤2||ν−|φ|||Cm(δ)→0,kask→0+withsuitablec0>0independentofk.Moreovertheaboveconvergencetozeroholdsfasterthananypowerofk.(c)Ask→0+,

(AJ0→0inL120)±→0,R;±andN(F12)±→±2πδz,(3.2.8)jj=1
21→πn2δ(A0)±zand(A0)±→±πnj(nj+1)δz,jjjkj∈Jj∈J 3.2PlanartopologicalChern–Simonsvortices81weaklyinthesenseofmeasureinR2.HereJ⊂{1,...,N}isasetofindicesidenti-fyingallofthedifferentvorticesinZ.Inparticular,1||(A220)±||L2(R2)→πnjand||(A0)±||L1(R2)→πnjnj+1.kj∈Jj∈Jiii)(Quantization)Thefollowingholdsrespectivelyforthemagneticflux,theelectricchargeandthetotalenergy:!Magneticflux=(F12)±=±2πN;R2!
ElectricchargeQ=J0=±2πkN;(3.2.9)R2±!E2TotalenergyE=±=2πνN.R2Theorem3.2.3abovesummarizesandimprovesseveraloftheresultsavailableinliteratureconcerningtopologicalChern–Simonsvortices(seee.g.,[Wa],[Y1],and[Ha3]).ItfurnishestheanalogousversionofTheorem3.2.1inthecontextofChern–Simonstheory.Infact,weobservethattheMaxwell–Higgsvortices(ofTheorem3.2.1)alsosatisfythe“concentration”property(3.2.8)asestablishedin[HJS].Itisbelievedthat,inanalogywithMaxwell–Higgsvortices,Chern–Simonstopo-logicalvortexconfigurationsareuniquelydeterminedoncethelocationoftheirvortexpointshasbeenspecified.Inotherwords,themaximalChern–Simonsvortexconfigu-rationdescribedaboveistheonlysolutionof(1.2.45)satisfying(3.2.2)and(3.2.5).Thisfacthasbeenestablishedforthecaseinwhichallvortexpointscoincide,namelyz1=z2=···=zN(see[Ha3]).Indeed,inthiscasethemagnitude|φ|2oftheHiggsfieldcanbeshowntoberadiallysymmetricaboutthe(multiple)vortexpoint(cf.[Ha3])anduniquenessfollowsfromageneralresultabouttheradialsolutionsestablishedbyChen–Hastings–McLeod–Yangin[CHMcLY].InSection3.3weshallprovethatuniquenessholdsforsmallvaluesoftheparame-terk>0(dependingontheassignedsetZofvortexpoints),providedthatallsolutionsof(1.2.45),(3.2.2),and(3.2.5)alsosatisfyuniform(exponential)decayestimates,asthoseclaimedin(3.2.6)and(3.2.7)forthe“maximal”solution.Recently,suchuniformboundshavebeenestablishedbyK.Choe[Cho],whodeducesauniquenessresultwhenk>0iseitherverysmallorverylarge(independenceofthesetZ).Seealso[ChN]forrelatedresults.Still,thequestionofuniquenessremainsopenforallvaluesofk>0andanyassignedsetZofvortexpoints.Alsowementionthatitisnotknownwhetherapointwiseestimateofthetype(3.2.3)isvalidforsolutionsoftheChern–Simonsselfdualequations(1.2.45)inR2.ToobtainTheorem3.2.3,weshallconstructasolutionufor(3.1.4)withtheappropriateasymptoticpropertiesatinfinity. 823PlanarSelfdualChern–SimonsVorticesTothispurpose,weshallsimultaneouslyaccountforthe(singular)behaviorofuatthevortexpointsandatinfinity,bytakinguoftheformu(z)=u0(z)+v(z)(3.2.10)withN|z−z2j|u0(z)=log.(3.2.11)1+|z−zj|2j=1Notethat,invirtueof(2.2.2),u0satisfies:N4N−u0=2−4πδzj.(3.2.12)j=11+|z−zj|2j=1Therefore,forthe(smooth)newunkownvtheproblemisreducedtosolve:N− v=λeu0+v1−eu0+v−4inR2(3.2.13)2j=11+|z−zj|2v(z)→0,as|z|→+∞.(3.2.14)Weset,N1g0(z)=42,(3.2.15)j=11+|z−zj|2and,notethatg0∈C∞(R2)∩Lp(R2),1≤p≤+∞.Soany(weak)solutionvof(3.2.13),(3.2.14)satisfiesv∈C∞(R2)∩L∞(R2).Weobservethefollowing:Proposition3.2.4Letvbea(smooth)solutionfor(3.2.13).Thenvsatisfies(3.2.14)ifandonlyif
v∈H1R2.(3.2.16)Furthermoreforu=u0+v(thesolutionof(3.1.4))thefollowingholds:i)
u∈LpR2∀p≥1,andu<0inR2;(3.2.17)ii)
!eu1−eu∈L1R2andλeu1−eu=4πN.(3.2.18)R2 3.2PlanartopologicalChern–Simonsvortices83Proof.Assumethatvsatisfies(3.2.14).Westarttocheckthatu=u0+v≤0inR2andfindthatthemorestrictinequalityclaimedin(3.2.17)followsfromtheHopflemma.Tothisend,wearguebycontradictionandsupposethatsupR2u>0.Sinceu→0as|z|→+∞,andu(z)→−∞asz→zjforeveryj=1,...N,weseethatuattainsits(positive)maximumvalueatapointz∗∈R2Zsuchthat,u∗:=u(z∗)>0andu(z∗)≤0.Butthisisimpossible,sinceby(3.1.4)weareleadtothecontradiction:0≤−u(zu∗1−eu∗<0;∗)=λeandsou<0inR2.(3.2.19)Nextweverifythat(3.2.18)holds.Tothispurpose,letχdenotethestandardcut-offfunctiondefinedbytheproperties:∞(R2),χ=1inB2χ∈C01(0),χ=0inRB2(0)and0≤χ≤1.(3.2.20)ThenforR>0,setχz)andusethisfunctionasatestfunctionin(3.2.13)R(z):=χ(Rtofind!!!v χu0+vu0+v−R=λe1−eχR−g0χR.(3.2.21)R2R2R2Observethat,asR→+∞,!!N!1g0χR→g0=42=4πN,(3.2.22)R2R2R21+|z−zj|2j=1andbymeansof(3.2.14),!!!v χR=v χR≤||v||L∞(R≤|z|≤2R)| χ|→0.R2{R≤|z|≤2R}{1≤|z|≤2}Thereforefrom(3.2.19)and(3.2.21)andthedominatedconvergencetheoremwede-'ducethat,eu0+v(1−eu0+v)∈L1(R2),andλ2eu0+v(1−eu0+v)=4πN.WeseeRnexthowtouse(3.2.18)toshowthatv∈H1(R2).Forδ>0sufficientlysmall,set2∪NBδ=Rj=1δ(zj),(3.2.23)sothatu∈L∞(δ).Forz∈δ,wecanusetheelementaryinequality:t|t||1−e|≥,∀t∈R(3.2.24)1+|t|toestimate
|u(z)|≤1+||u||∞|1−eu(z)|≤1+||u||∞e||u||L∞(δ)eu(z)1−eu(z).L(δ)L(δ) 843PlanarSelfdualChern–SimonsVorticesThus,from(3.2.18),wemayactuallyconcludethatu∈L1(δ)∩L∞(δ).Ontheotherhand,since|u|≤|u0|+|v|∈Lp(Bδ(zj)),∀p≥1,∀j=1,...,Nweseethat
u∈LpR2,∀p≥1.(3.2.25)Noticethatforu0in(3.2.11)wehave

u12∩LqR2,∀q>1.(3.2.26)0∈LlocRTherefore,bycombining(3.2.25)and(3.2.26)weobtain,
v∈LqR2,∀10suchthat,sup|v|≤C||v||+||f||,∀z∈R2.L2(B2(z))L2(B2(z))B1(z)Consequently,forR>1,sup|v|≤C||v||L2(|z|≥R−1)+||f||L2(|z|≥R−1)→0,asR→+∞|z|≥R+1and(3.2.14)isestablished.Remark3.2.5Wewishtoobservethatthemereassumptionp({|z|≥ru=u0+v∈L0}),forsomep≥1andr0>0sufficestoensurethatasolutionvof(3.2.13)alsosatisfies(3.2.14).Thisfollowseasilyfromtheobservationthatu+=max{u,0}andu−=max{−u,0}definesubharmonicfunctionsinR2Z.Thus,bythemeanvaluetheoremwededucethat:u(z)→0,as|z|→+∞.Inthesameway,thecondition∞({|z|≥ru=u0+v∈L0}),forsomer0>0,(3.2.29)impliesu(z)→0,as|z|→+∞,whenevervsatisfies(3.2.13)and
eu0+v1−eu0+v∈L1R2.(3.2.30)Indeed,wecanuse,asabove,theinequality(3.2.24)togetherwith(3.2.28)and(3.2.30),toseethatu0+v∈L1({|z|≥r0}),fromwhich(3.2.14)follows.Noticethatifweassume(3.2.30),thenwemaystilldeduce(3.2.14)undertheweakercondition:(u0+v)−∈Lp({|z|≥r0})forsome1≤p≤+∞andr0>0.Indeed(3.2.30)allowustocontrolthepositivepart(u0+v)+bymeansoftheestimate:++
0≤(u0+v)+≤e(u0+v)1−e(u0+v)≤eu0+v1−eu0+v∈L1R2.Next,atfixedλ>0,weprovidesomeusefulexponentialdecayestimatesatinfinityforsolutionsof(3.2.13),(3.2.14).Proposition3.2.6Letvbea(classical)solutionfor(3.2.13),(3.2.14),andsetu=u0+v.Foreveryε∈(0,1)andδ>0,thereexistsuitableconstantsCε=Cε(λ)>0andCε,δ=Cε,δ(λ)>0,suchthat√(i)0<1−eu(z)≤Cεe−(1−ε)λ|z|inR2;(3.2.31)√(ii)|u(z)|+|∇u(z)|≤Cε,δe−(1−ε)λ|z|,∀z∈δ=R2∪NBδ(zj).j=1 863PlanarSelfdualChern–SimonsVorticesProof.Recallthatfin(3.2.28)satisfies||f||L1(R2)≤8πN,||f||L∞(R2)≤λ+4N,(3.2.32)andsincevisbounded,wecanuseGreen’srepresentationformulatowrite!1z−y∇v(z)=f(y).22πR2|z−y|Consequently,foreveryR>0weestimate:!!111|∇v(z)|≤|f(y)|+|f(y)|2π{|z−y|>R}|y−z|{|z−y|r0,|z−zj|1+|z−zj|2|z|3j=1andwiththehelpof(3.2.32)and(3.2.33),wederive*1λ|∇(u0+v)(z)|≤8N+1+,∀|z|>r0.(3.2.34)|z|34NToobtain(3.2.31),weintroducethefunctionψ(z)=eA(R−|z|)+eu0+v−1,withR>r0andAtobespecifiedbelow.Clearly,ψ≥0inBR.Toanalyzewhathap-pensfor|z|≥R,weobservethatψsatisfiesthefollowingboundaryvalueproblem:⎧
⎪⎨ ψ=A2−A−λe2(u0+v)eA(R−|z|)+λe2(u0+v)ψ+eu0+v|∇(u0+v)|2|z|(3.2.35)⎪⎩ψ|∂Bandψ(z)→0,as|z|→+∞.RTherefore,ifwesupposethatinfψ<0,thenwefindzR∈R2suchthat|zR|>R|z|≥Randψ(zR)=infψ<0.Inparticular,∇ψ(zR)=0leadstotheidentity|z|≥R 3.2PlanartopologicalChern–Simonsvortices87e(u0+v)(zR)|∇(u0+v)(zR)|2=−AeA(R−|zR|)zR,∇(u0+v)(zR),(3.2.36)|zR|and ψ(zR)≥0.Using(3.2.34)into(3.2.36),wederivetheestimate*e(u0+v)(zR)|∇(u0+v)(zR)|2≤AeA(R−|zR|)8N1+1+λ,|z|34Nwhichwecaninsertinto(3.2.35)tofind0≤ ψ(zR)+≤A2−A−λe2(u0+v)(zR)+8NA1+1+λeA(R−|zR|).(3.2.37)|zR||zR|34N,+Noticethat,A2λ0:=16N+(4N+1)λ−4N1+satisfies4N*2λA0+8NA01+=λ;4NandforeveryA∈(0,A0)thereexistsauniqueδ∈(0,1),*2λA+8NA1+=λ(1−δ).4NRecallingthateu0+v→1as|z|→+∞,wereachacontradictionin(3.2.37)withasufficientlylargeR>0satisfying,R2>8Nandinfe2(u0+v)>1−δ.|z|≥RInotherwords,wehaveestablishedthat,∀A∈(0,A0)thereexistsaconstantCA>0suchthat,(3.2.38)1−e(u0+v)(z)≤CAe−A|z|,inR2.+SinceAλ,theestimate(3.2.38)comesupshortincoveringtheclaimedes-0<4N√timate(3.2.31).Weshownext,that(3.2.38)actuallyholdswithA=(1−ε)λ,∀ε∈(0,1).Tothispurpose,observethatfrom(3.2.38)wehave
(u0+v)(z)(u0+v)(z)−A|z|,∀z∈|(u0+v)(z)|+λe1−e≤Cδeδ(3.2.39)foreveryA∈(0,A0)andasuitableconstantCδ>0,whichnowdependsalsoonδ>0. 883PlanarSelfdualChern–SimonsVorticesSince− (u0+v)=λeu0+v(1−eu0+v)inδ,wecanuse(3.2.39)togetherwiththewell-knowngradientestimatesforPoisson’sequation(e.g.,seeTheorem3.9in[GT]),toobtain:−A|z|,∀z∈2N|∇(u0+v)(z)|≤Ce2δ=R∪j=1B2δ(zj).(3.2.40)Butifwesubstitute(3.2.40)into(3.2.36),wecanimprove(3.2.37);andforR>0sufficientlylarge,wecanseethatinfψ<0implies:A2−λinfe2(u0+v)≥0.|z|≥R|z|≥R√SinceforanyA∈(0,λ)wecaneasilycontradictthelatteroftheinequalitiesabovebylettingR→+∞,wearriveatthedesiredestimateof√(i)in(3.2.31).Inturn,for∀ε∈(0,1)wecantakeA=(1−ε)λin(3.2.39),andconsequentlyin(3.2.40),andthusobtainestimate(ii)in(3.2.31).Wenowturntoconstructasolutionfor(3.2.13),(3.2.14).OnthebasisofProposition3.2.4,weknowthatH1(R2)furnishestheappropriatefunctionalspaceinwhichtosearchforthesolution.Infact,inH1(R2)wecanformulateourprobleminavariationalformbyconsid-eringthefunctional,!!!
I12λu0+v212λ(v)=|∇v|+e−1+g0v,v∈HR.2R22R2R2ByvirtueofLemma2.4.7,weknowthatIλ∈C1(H1(R2)).Moreover,onthebasisoftheellipticregularitytheory,anycriticalpointofIλinH1(R2)furnishesaclassicalsolutionfor(3.2.13),whichbyProposition3.2.4,alsosatisfies(3.2.14).Thus,weonlyneedtofocusonthesearchforcriticalpointsforIλinH1(R2).Inthisdirectionweobtain:Proposition3.2.7Forλ>8||g0||222thefunctionalIλisboundedfrombelowandL(R)coercieveinH1(R2).Itattainsaninfimumatvλ∈H1(R2)thatsatisfies:||u0+vλ||L2(R2)→0,asλ→+∞.(3.2.41)Proof.Wecanusetheelementaryinequality(3.2.24)toestimate!!2!2!1−eu0+v2≥(u0+v)≥1v−g|u20|,22R2R2(1+|(u0+v)|)2R2(1+|v|)R2andfindforanyv∈H1(R2),!212λvgλ222Iλ(v)≥||∇v||L2(R2)+2−||u0||L2(R2)−||g0||L2(R2)||v||L2(R2).24R21+|v|22(3.2.42) 3.2PlanartopologicalChern–Simonsvortices89Ontheotherhand,bymeansofSobolev’swell-knowninterpolationinequality,!!!
v4≤2v2|∇v|2,v∈H1R2,R2R2R2weobtain!2!!v2v2≤v2(1+|v|)22R2R2(1+|v|)R2!!
v2≤2v21+2||∇v||22L2(R2)R2(1+|v|)R2andarriveattheinequality!1v22
122||v||L2(R2)≤221+2||∇v||L2(R2).(3.2.43)R2(1+|v|)Inviewof(3.2.42),thisimplies:!212λvIλ(v)≥||∇v||L2(R2)+224R2(1+|v|)!1v2
2−||g220||L2(R2)221+2||∇v||L2(R2)−gλ||u0||L2(R2)R2(1+|v|)!2
12λvε2≥||∇v||L2(R2)+2−1+2||∇v||L2(R2)24R2(1+|v|)2!212v2−||g0||L2(R2)2−gλ||u0||L2(R2)εR2(1+|v|)2!12λ||g0||L2(R2)v2ε=−ε||∇v||L2(R2)+−2−24εR2(1+|v|)2−gλ||u2,0||L2(R2)forv∈H1(R2)andforsufficientlysmallε>0.Therefore,forλ>8||g0||222weL(R)||g0||21λL2(R2)mayfixελ>0suchthatδλ=−ελ>0andσλ=−>0.Weobtain24ελ!2
2v12Iλ(v)≥δλ||∇v||L2(R2)+σλ2−Cλ,∀v∈HR,(3.2.44)R2(1+|v|)forsuitableCλ>0. 903PlanarSelfdualChern–SimonsVorticesInthisway,wehaveestablishedthatIλisboundedfrombelowinH1(R2),pro-videdλ>8||g0||222.L(R)
1Denoteby||v||=||∇v||2+||v||22thenorminH1(R2)using(3.2.43)intoL2L2(3.2.44),wefind||v||22σλL2(R2)Iλ(v)≥δλ||∇v||L2(R2)+22−Cλ1+2||∇v||L2(R2)σλ||v||222+||∇v||2222L(R)L(R)=δλ||∇v||L2(R2)+22−Cλ1+2||∇v||L2(R2)*1≥δλσλ(1+2||v||2)−βλ2forsuitableβλ>0;andthisprovesthecoercivenessofIλ.SinceIλisweaklylowersemicontinuousinH1(R2),weconcludethat,forλ>8||g0||222,thefunctionalIλattainsaninfimumatapointvλ∈H1(R2).L(R)Itremainstoestablish(3.2.41).Claim:Thereexistsuitableconstantsa0,b0>0suchthat∀λ>8||g0||222weL(R)haveIλ(vλ)=infIλ≤a0logλ+b0.(3.2.45)H1(R2)Toobtain(3.2.45),weshallevaluateIλoverasuitabletestfunction.Tosimplifynota-tion,wetakethevortexpointsz1,...,zNtobedistinct,sincewiththeobviousmodi-ficationswecantreatalsothecaseofmultiple-vortexpoints.Letε>0butsufficientlysmall(tobespecifiedbelow),suchthattheballsBε(zj)aremutuallydistinct.Set⎧⎨−u0(z)z∈R2∪NBε(zj)j=1vε(z)=
2|z−z|2(3.2.46)⎩−logε+logkz∈Bε(zj),j=1,...,N.1+ε2k=j1+|z−zk|2Sothatvε∈H1(R2)and,⎧⎪⎨0z∈R2∪NBε(zj)j=1u0(z)+vε(z)=⎪⎩z−zj21+ε2(3.2.47)logε1+|z−zj|2z∈Bε(zj),j=1,...,N. 3.2PlanartopologicalChern–Simonsvortices91Weestimate,!!!I12λu0+vε2λ(vε)=|∇vε|+(1−e)+g0vε2R22R2R2N!1≤cN{|z−z|≥ε}|z−zj|2(1+|z−zj|2)2j=1j!22λz−zj1+ε2+1−2{|z−z|<ε}ε1+|z−zj|2j!z−zj21+ε2+g0(z)log+C1(3.2.48){|z−z|<ε}ε1+|z−zj|2j⎛!!221z21+ε≤C2⎝+λ1−{|z|>ε}|z|2(1+|z|2)2{|z|<ε}ε1+|z|2⎞!z2+log+1⎠{|z|<ε}ε⎛⎞!2!12121+ε2211=C3⎝log+λεr1−rdr+εrlogdr+1⎠,ε01+ε2r0r2withsuitablepositiveconstantscN,C1,C2andC3thatareindependentofλandε.Consequently,bychoosingε2=1,weconclude(3.2.45).λAtthispoint,wecanuse(3.2.45)todeducetheestimates||∇v2λ||L2(R2)≤c0(logλ+||u0+vλ||L2(R2)+1),(3.2.49)!2!(u0+vλ)≤(1−eu0+vλ)22R2(1+|u0+vλ|)R2c0≤logλ+||u0+vλ||L2(R2)+1(3.2.50)λforasuitableconstantc0>0andindependentofλ.Consequently,!2(u20+vλ)R2!!(u0+vλ)222≤(u0+vλ)(1+|u0+vλ|)(3.2.51)2R2(1+|u0+vλ|)R2!!(u0+vλ)2244≤2||u0+vλ||L2(R2)+8||u0||L4(R2)+8vλ,R2(1+|u0+vλ|)R2 923PlanarSelfdualChern–SimonsVorticesand!v4≤2||v22λλ||L2(R2)||∇vλ||L2(R2)R2(3.2.52)≤4||u22220+vλ||L2(R2)||∇vλ||L2(R2)+||u0||L2(R2)||∇vλ||L2(R2).So,wecaninsert(3.2.52)into(3.2.51),tofindasuitableconstantC>0(andinde-pendentofλ)suchthat,||u40+vλ||L2(R2)!(3.2.53)2

|u0+vλ|22≤C2||u0+vλ||L2(R2)+1||∇vλ||L2(R2)+1.R21+|u0+vλ|Thus,substituting(3.2.49)and(3.2.50)into(3.2.53)weseethat,||u0+vλ||L2(R2)→0,asλ→+∞.Corollary3.2.8Foreveryλ>0,problem(3.1.4),(3.1.9)admitsamaximalsolutionuλ<0,monotoneincreasingwithrespecttotheparameterλ>0,andsatisfying:||uλ||L2(R2)→0,asλ→+∞.(3.2.54)Proof.Recallthatbymaximalwemeanthatuλsatisfiesu2λ(z)≥u(z),∀z∈RZ,foranysolutionuof(3.1.4),(3.1.9).Letusfirsttreatthecasewhereλ>8||g0||222(asrequiredinProposition3.2.7),L(R)andset,1(R2)solves(3.2.13),(3.2.14)}≤−uv−(z)=sup{v(z):v∈H0(z).So,v−∈H1(R2)definesa(bounded)subsolutionfor(3.2.13),(3.2.14),whichyieldstotheexistenceofasolutionvˆλ∈H1(R2)for(3.2.13),(3.2.14)satifying:vˆλ≥v−inR2(seee.g.,Theorem2.4in[St1]).Consequently,uλ=u0+ˆvλdefinesamaximalsolutionfor(3.1.4),(3.1.9)andinparticular,0>u2λ(z)≥(u0+vλ)(z),a.e.inRwithvλgiveninProposition3.2.7.Therefore,||uλ||L2(R2)≤||u0+vλ||L2(R2)→0,asλ→+∞.For0<λ1<λ2,thefunctionvˆλdefinesastrictsubsolutionfor(3.2.13).Con-1sequently,for(3.2.14)withλ=λ2,wededucethenecessarycondition:vˆλ(z)<1vˆλ(z)∀z∈R2;andthedesiredconclusionfollowsinthiscase.2 3.2PlanartopologicalChern–Simonsvortices93Nowtoeliminatetheconditionλ>8||g0||222,weobservethatforaL(R)givenε>0,nothingchangesinouranalysisifwereplaceu0in(3.2.11)withN|z−zj|2u0,ε(z)=j=1log1+|z−z|2;thusg0(z)in(3.2.15)becomesg0,ε(z)=εj
−2N1+|z−z24j=1εj|.Thereforetheaboveconclusionholds,providedthatλ>8||g0,ε||222=O(ε2).Wecanalwaysverifythisconditionbychoosingε>0L(R)sufficientlysmall.Wementionthatthefirstexistenceresultfor(3.1.4),(3.1.9)wasestablishedin[Wa],whilewereferto[Y1]foranalternativeproofbasedonaniterationscheme.Property(3.2.54)hastheinterestingconsequenceofimplyingastrong“localiza-tion”propertyforthesolutionuλaroundthevortexpoints(asλ→+∞),asisconsis-tentwithphysicalapplications.Morepreciselythefollowingholds:Proposition3.2.9Letp0≥1andλ0≥0.Assumethatwehaveasolutionvλof(3.2.13)suchthatuλ=u0+vλ(solutionof(3.1.4))satisfies:p02uλ∈L(R)and||uλ||Lp0(R2)≤C,∀λ≥λ0withasuitableC>0(independentofλ).Then(i)foreveryε∈(0,1)andδ>0theestimatesof(3.2.31)holdwithuniformconstantsCεandCε,δindependentofλ≥λ0.(ii)Asλ→+∞:||uλ||Lp(R2)→0,∀p≥1;(3.2.55)Nλeuλ(1−euλ)→4πδ2z,weaklyinthesenseofthemeasureinR.(3.2.56)jj=1Furthermore,foranyε∈(0,1)thereexistsaconstantλε≥λ0,suchthat∀λ≥λεthefollowinguniformestimateshold:(iii)
√uλ(z)(1−ε)λ(R0−|z|),∀|z|≥R|∇uλ(z)|≤λc01−e≤C0e0;(3.2.57)withsuitableconstantsR0>0,c0>0,andC0>0thatareindependentofλandε.(iv)sλ||uλ||Cm(δ)→0,asλ→+∞,(3.2.58)foreverys,m∈Z+andδ>0,whereδ=R2∪NBδ(zj).j=1Proof.Firstofall,observethatbyProposition3.2.4andRemark3.2.5weknowthat!uuλ<1inR2andλeuλ(1−euλ)=4πN.(3.2.59)λ→0as|z|→+∞,eR2 943PlanarSelfdualChern–SimonsVorticesHence,uλdefinesasuperharmonicfunctioninR2{z1,...,zN}.Thus,foreveryδ>0andz∈δ=R2∪NBδ(zj),wecanusethemeanvaluetheoremtogetj=1!1|uλ(z)|=−uλ(z)≤|uλ|.(3.2.60)πδ2Bδ(z)Consequently,11p0C||uλ(z)||L∞(2δ)≤2||uλ||Lp0(δ)≤2.(3.2.61)πδπδBymeansof(3.2.24),(3.2.59),and(3.2.61),foranyp≥1wefind,!!ppuλp|uλ|≤(1+||uλ||L∞(δ))(1−e)δδ!≤(1+||uλ||∞)pe||uλ||L∞(δ)euλ(1−euλ)≤cδ,p(3.2.62)L(δ)δλwithasuitableconstantcδ,p>0dependingonδandpbutindependentofλ.Inpar-ticular,using(3.2.62)withp=1and(3.2.60),wecanimprove(3.2.61)andconcludeCδ||uλ||L1(δ)+||uλ||L∞(δ)≤,(3.2.63)λandconsequentlyobtainuλuλλ||e(1−e)||L∞(δ)≤Cδ(3.2.64)withasuitableconstantCδ>0thatisindependentofλ.Toanalyzewhathappensaroundthevortexpoints,weobservethatvλadmitsp02L(R)-normuniformlyboundedinλ.Alsosince vλisuniformlyboundedinlocL1(R2)and vλ≤g0∈L1(R2)∩L∞(R2),wecanusewell-knownellipticesti-matestoensurethatvλislocallyanduniformlyboundedfrombelow.Inparticular,for∀δ>0wefindaconstantdδ>0andNB|uλ|=−uλ≤|u0|+dδ,inDδ:=∪j=1δ(zj).Ontheotherhand,by(3.2.63)weknowthatuλ→0pointwisea.e.,asλ→+∞.So,bydominatedconvergence,foranyp≥1:||uλ||Lp(Dδ)→0asλ→+∞,andthistogetherwith(3.2.63)yieldsto(3.2.55).Furthermore,observethatfrom(3.2.63)and(3.2.64)wealsoknowthatbothvλand vλareboundedinL∞(δ)anduniformlyinλ.Therefore,thewell-knownel-lipticandSobolev’sestimatesimplythat|∇vλ|isalsouniformlyboundedinL∞(δ),andforsuitablepositiveconstantsc1andr1,thefollowingholds:c18Nc1|uλ(z)|≤and|∇uλ(z)|≤+,∀|z|>R,(3.2.65)λR2|z|3R2 3.2PlanartopologicalChern–Simonsvortices95√foreveryR≥r1.LetR0:=max{r1,8N,|zj|j=1,...,N}+1,andforε∈(0,1),R≥R0considerthefunctionC1√ψe(1−ε)λ(R−|z|)+eu0+vλ−1λ(z)=λwithC1choosen(dependingonlyonR)insuchawaythat,ψλ≥0in∂BR.Noticethatwecanalwaysattainthepropertyabovebyvirtueof(3.2.63).Sinceψλ(z)→0as|z|→+∞,wecanapplytoψλtheargumentsgivenintheproofofProposition3.2.6,andusing(3.2.65)concludethat,2c12uλifinfψλ<0,thennecessarily(1−ε)+√(1−ε)−infe≥0.|z|≥RR2λ|z|≥ROnthebasisof(3.2.63)and(3.2.65),theinequalityaboveiscertainlyviolatedeitherforafixedλ≥λ0andforR=Rε>0choosensufficientlylarge(accordingtoε),orforR=R0andforλsufficientlylarge(accordingtoε).Inthefirstcase,wededucetheuniformestimatesasclaimedin(i).Inthesecondcase,wefindλε>0suchthat∀λ≥λε,C1√uλ(z)≤(1−ε)λ(R0−|z|),∀|z|≥R0<1−ee0.(3.2.66)λConsequently,∀z:|z|≥R0wehave|uuλ(1−euλ)≤(1+λ)(1−euλ)λ(z)|+|uλ(z)|=|uλ(z)|+λe√≤C(1−ε)λ(R0−|z|)0ewithsuitableC0>0thatisindependentofλandε.Thus,usingthewell-knowngradientestimatesforPoisson’sequationweconclude(3.2.57).From(3.2.66)wealsohave!!√u|z|λuλ−(1−ε)λλe(1−e)≤C0e2{|z|≥2R0}{|z|≥2R0}Cε√≤−(1−ε)λR0,λ≥λ√eε(3.2.67)λforanygivenε∈(0,1)andacorrespondingsuitableconstantCε>0thatdependsonεonly.Inparticular,!λeuλ(1−euλ)→0asλ→+∞.(3.2.68){|z|≥2R0}Ontheotherhand,∀ϕ∈C∞(R2),wefindthat0!N!!λeuλ(1−euλ)ϕ−4πϕ(zj)=−uλϕ=−uλ ϕ→0,(3.2.69)R2R2R2j=1 963PlanarSelfdualChern–SimonsVorticesasλ→+∞,andthat(3.2.56)followsfrom(3.2.68)and(3.2.69).Inaddition,ifwetakein(3.2.69)afunctionϕ∈C∞(B3R)suchthatϕ=0in∪NBδ(zj)whileϕ=100j=12inδ∩B2R,thenweobtain0!euλ(1−euλ)≤cλδ||uλ||L1(δ)(3.2.70)δ∩B2R02withcδ>0asuitableconstantdependingonδ>0only.Wecanuse(3.2.67)and(3.2.70)toimproveinequalities(3.2.63)and(3.2.64).Indeed,arguingasin(3.2.62)wefind!!≤Cuλuλuλuλ||uλ||L1(δ)1,δe(1−e)+e(1−e)δ∩B2R0{|z|≥2R0}C2,δCε,δ−(1−ε)√λR1≤||uλ||L1(+√e0≤Cδ,λδ)λ22λwherewehaveused(3.2.63)toderivethelastinequalitywithasuitableconstantCδ>0.Iteratingtheargumentabove,weseehowtoestimatetheL1(δ)-normofuλ(and∞(1henceitsLδ)-norm),foranypowerof.Inotherwords,forλsufficientlylarge,λwehaveuλuλC||uλ||L1(δ)+||uλ||L∞(δ)+||λe(1−e)||L∞(δ)≤s,(3.2.71)λforanys≥1,andasuitablepositiveconstantC=C(δ,s)independentofλ.Recallthat−uλ=λeuλ(1−euλ)inδ.Sobywell-knowngradientestimatesforPoisson’sequation(cf.[GT])from(3.2.71),followsananalogousL∞-estimatefor|∇uλ|inδ.Atthispoint,byfamiliarbootstraparguments,wearriveattheestimateC||uλ||Cm(δ)≤s,λ>0large;(3.2.72)λforanys≥1andasuitableconstantC=C(δ,s,m)>0independentofλ.Clearly,(3.2.58)followsfrom(3.2.72),asλ→+∞.TheresultsofProposition3.2.7and3.2.9canbesummarizedasfollows:Corollary3.2.10Forλ>0,problem(3.1.4)admitsamaximalsolutionuλmonoton-icallyincreasinginλ>0,suchthat,i)Topologicalbehavior:!2,uuλuλ(a)uλ<0inRλ(z)→0as|z|→+∞,andλe(1−e)=4πN.R2(b)Foreveryλ>0,ε∈(0,1),andδ>0,thereexistcontantsCε>0andCε,δ>0(independentofλ)suchthatforλ≥λ0wehave 3.2PlanartopologicalChern–Simonsvortices97√0<(1−euλ)≤C−(1−ε)λ|z|2εe,∀z∈R,(3.2.73)and√|u−(1−ε)λ|z|λ(z)|+|∇uλ(z)|≤Cε,δe,∀z∈δ.(3.2.74)ii)Asymptoticbehaviorasλ→+∞:uλsatisfiesproperties(3.2.55)–(3.2.58)ofProposition3.2.9.Furthermore,λ(1−euλ)2→4πn2δuλzandλ(1−e)→4πnj(nj+1)δz,(3.2.75)jjjj∈Jj∈JweaklyinthesenseofmeasureinR2.Inparticular,!!(1−euλ)2→4πn2andλ(1−euλ)→4πnλjj(nj+1),R2R2j∈Jj∈JwhereJ⊂{1,...,N}isasetofindicesidentifyingalldistinctpointsin{z1,...,zN}andnjisthemultiplicityofzj,forj∈J.Proof.Itremainsonlytoshow(3.2.75),whichweshallestablishbyprovingthat,∀δ>0small,!λ(1−euλ)2→4πn2,asλ→+∞.jBδ(zj)Then,(3.2.75)followsbymeansofproperties(3.2.56)–(3.2.58)andtheidentity:1−euλ=(1−euλ)2+euλ(1−euλ).Forsimplicityandwithoutlossofgenerality,wetakezj=0.Notethat,forδ>0small,wehavethatuλ(z)=2njlog|z|+(smoothfunctioninBδ(0));andtherefore,z·∇uλ(z)=2nj+ϕλ(z),withϕλsmooth(aroundtheorigin)andϕλ(0)=0.Hence,wecanobtainaPohozaev-typeidentity(see(5.2.14))bymultiplingtheequationbyz·∇uλ(z)andintegratingoverBδ(0),asfollows:!!−uuλuλ2λ∇uλ(z)·z=λe(1−e)∇uλ·z−8πnj|z|<δ|z|<δ!
!λuλ2uλ22=−divz(1−e)+λ(1−e)−8πnj2|z|<δ|z|<δ!!λuλ2uλ22=−δ(1−e)dσ+λ(1−e)−8πnj.2|z|=δ|z|<δ(3.2.76) 983PlanarSelfdualChern–SimonsVorticesUsingtheuniformestimatesin(3.2.58)weseethat,!λ(1−euλ)2dσ=o(1),asλ→+∞.(3.2.77)|z|=δNext,forwhatconcernstheleft-handsideof(3.2.76),denotebyn=ztheoutward|z|normaloftheball{z:|z|≤δ}.Takingintoaccountthesingularityof∇uλatzj,wehave!−uλ∇uλ(z)·z|z|≤δ!!
12=−div(∇uλ∇uλ(z)·z)+divz|∇uλ||z|≤δ2|z|≤δ!!122=−(∇uλ·z)(∇uλ·n)dσ+|∇uλ|n·zdσ−4πnj|z|=δ2|z|=δ!212∂uλ22=δ|∇uλ|−dσ−4πnj→−4πnj,asλ→+∞,|z|=δ2∂n(3.2.78)asfollowsfrom(3.2.58).Hencepassingtothelimitasλ→+∞in(3.2.76),from(3.2.77)and(3.2.78),weconcludethat!λ(1−euλ)2=4πn2+o(1),asλ→+∞;j|z|≤δand(3.2.75)follows.Remark3.2.11Noticethat,moregenerally,property(3.2.75)remainsvalidforanyfamilyofsolutionsuλsatisfyingtheassumptionsofProposition3.2.9.ProofofTheorem3.2.3.Letusfirstestablishthedesiredstatementwhenthesymmetrybreakingparameterν=1.Fork>0,setλ=4;thenitsufficestosubstitutethek2maximalsolutionuλofCorollary3.2.10into(2.1.8),(2.1.9)and(2.1.13)toobtainφ±and(Aα)±,α=0,1,2whichdefinesasolutionfortheChern–Simonsselfdualequations(1.2.45),with|φ±|2=euλ.Therefore,using(3.1.12)(withu=uλ)wecheckthevalidityofproperties(i),(ii)and(iii)directlybymeansofthepropertiesofuλasclaimedinCorollary3.2.10(recallthatν=1inthiscase).Inparticular,from(3.2.59)weseethat,!!
!(F012)±=±2πNandJ=k(F12)±=±2πkN.R2R2±R2 3.3Auniquenessresult99Toevaluatethetotalenergy,weobservethatforR>0,!!∂±k (A0)±=±k(A0)±dσ{|z|≤R}{|z|=R}∂n!z=∇(1−euλ)·dσ{|z|=R}|z|!z=euλ∇uλdσ{|z|=R}|z|≤2πRmax|∇uλ|→0,asR→+∞;|z|=Rasitfollowsfromthegradientestimatesofuλin(3.2.57).Therefore, (A0)±∈'L1(R2)and2 (A0)±=0.Hence,from(3.1.13)weconcludethat,R!!E=±F12=2πN,R2R2andthiscompletestheproofofTheorem3.2.3whenν=1.Totreatgeneralvaluesoftheparameterν,wesimplytakethesolutionuλofCorollary3.2.10withvortexpointslocatedatν2zj,j=1,...,N.Thedesiredvortexconfigurationisconstructedasabovebyusingthescaledfunctionu2z)+2logν,λ(νwhichsatisfies(3.1.1)togetherwithallthedesiredproperties.3.3AuniquenessresultWeconcludetheanalysisof(3.1.4),(3.1.9)bydiscussinguniquenessoftopologicalsolutions.Westarttomentionthatuniquenessholdswhenallvortexpointscoincide,saywiththeorigin.IfwedenotebyN∈Nthevortexmultiplicity,thenthissituationisdescribedbytheproblem−u=eu(1−eu)−4πNδz=0(3.3.1)u(z)→0as|z|→+∞,√aswecanscaleouttheparameterλbythechangeofvariablesz→λz.OnthebasisofaresultofHan[Ha3](seealso[CFL]),weknowthat(3.3.1)admitsauniquesolutionuradiallysymmetricabouttheorigin.Moreover,accordingtotheresultsestablishedintheprevioussection,wealsoknowthatusatisfies!u∈Lp(R2)∀p≥1,u<0inR2andeu(1−eu)=4πN,(3.3.2)R2 1003PlanarSelfdualChern–SimonsVorticesand∀ε∈(0,1),thereexistsaconstantCε>0suchthat
u(z)≤C−(1−ε)|z|,∀|z|≥1.(3.3.3)|u(z)|+|∇u(z)|+1−eεeFurthermore,wecanusePohozaev’sidentity(asin(3.2.76)–(3.2.78))togetherwiththeestimatesabovetofind:!!(1−eu)2=4πN2and(1−eu)=4πN(1+N).(3.3.4)R2R2Weshowthatuisnon-degenerateinthefollowingsense:Theorem3.3.12Letubetheuniqueradiallysymmetricsolutionfor(3.3.1).Thenthe(linearized)problem− ϕ+eu(2eu−1)ϕ=0(3.3.5)ϕ∈H1(R2),admitsonlythetrivialsolutionϕ=0.Proof.ArguingasinProposition3.2.7,forε0>0sufficientlysmallwecandecomposer212u=u0+vinsuchawaythatu0(r)=Nlog22andv∈H(R)correspondtotheε+r0(global)minimumforthefunctional,!!!121u0+v2I(v)=|∇v|+e−1+g0v,2R22R2R24πε2withg0.Consequently,0(z)=g0(|z|)=(ε2+|z|2)20!!
|∇ϕ|2+eu(2eu−1)ϕ2=I(v)ϕ,ϕ≥0,ϕ∈H1R2.(3.3.6)R2R2So,anypossiblenon-trivialsolutionfor(3.3.5)wouldcorrespondtothefirsteigen-functionrelativetothe(first)eigenvalue(equaltozero)forthelinearoperator:L=−+eu(2eu−1)inH1(R2).Sinceuisradiallysymmetric,thennecessar-ilyalsothefirsteigenfuctionϕmustberadiallysymmetric.Therefore,ifwearguebycontradictionandletϕ=0beasolutionfor(3.3.5)thenϕ=ϕ(r)andwithoutlossofgenerality,wecanalsoassumeϕ>0.Thus,summarizingthepropertiesabove,wehavethatu=u(r)satisfies⎧⎪⎪u¨(r)+1u˙(r)=eu(r)(eu(r)−1),∀r>0⎨ru(r)=2Nlogr+O(1),asr→0+(3.3.7)⎪⎪⎩0<1−eu(r)≤C−(1−ε)rεe,∀r>0withε∈(0,1)andCε>0asuitableconstantdependingonεonly(see(3.2.31)). 3.3Auniquenessresult101Whileforϕ=ϕ(r)wehave:⎧⎪⎨ϕ¨+1ϕ˙=eu(r)(2eu(r)−1)ϕ(r)rϕ>0(3.3.8)⎪⎩'+∞22(ϕ˙+ϕ)rdr<+∞.0Wemakeaconvenientchangeofvariablesandset,U(t)=u(et)andψ(t)=ϕ(et).(3.3.9)Properties(3.3.7)readasU¨=e2teU(t)eU(t)−1inR(3.3.10)U(t)=2Nt+O(1)ast→−∞,(3.3.11)tU(t)<0inRand1−eU(t)≤Cεe−(1−ε)e,ast→+∞(3.3.12)forε∈(0,1)andCε>0asuitableconstantdependingonεonly.HenceU¨<0inR2andsoU˙ismonotonedecreasinginRandast→±∞wefindU(t)limU˙(t)=lim=2N,(3.3.13)t→−∞t→−∞tasfollowsfrom(3.3.11).Moreover,U(t)limU˙(t)=lim=0,(3.3.14)t→+∞t→+∞tasfollowsfrom(3.3.12).Whilefrom(3.3.8)wehaveψ(¨t)=e2teU(t)2eU(t)−1ψinR'+∞ψ˙222t(3.3.15)ψ>0,inRand+ψedt<+∞.−∞Hence,ψ<¨0fort→−∞whileψ>¨0fort→+∞.Thereforeψ˙isamonotonefunctionfor|t|large,soitadmitsalimitfort→±∞,whichmustvanishbytheintegrabilitypropertyin(3.3.15).Namely,ψ(t)limψ(˙t)=0=lim.(3.3.16)t→±∞t→±∞tAsaconsequencewefindthatψ<˙0ast→±∞,soψisstrictlymonotonedecreas-ingfort→±∞,andagainby(3.3.15)wederivelimψ(t)=0.(3.3.17)t→+∞Now,wemultiply(3.3.15)byU˙andfind:

ψ¨U˙=e2tU(t)U(t)U˙(t)=e2tdU(t)U(t)ψ(t)e2e−1ψ(t)ee−1.dt 1023PlanarSelfdualChern–SimonsVorticesThatis:


dd2tU(t)U(t)d2tU(t)U(t)(ψ˙U˙)−ψ˙U¨=eψ(t)ee−1−eψ(t)ee−1dtdtdt


d2tU(t)U(t)2tU(t)U(t)=eψ(t)ee−1−2eψ(t)ee−1dt
−e2teU(t)eU(t)−1ψ.˙Therefore,ifweuse(3.3.10)weobtain:

dψ(˙t)U˙(t)+e2tU(t)U(t)2tU(t)U(t)ψ(t)e1−e=2eψ(t)e1−e.(3.3.18)dtOntheotherhand,by(3.3.12),(3.3.14),(3.3.16),and(3.3.17)wehave:

limψ(˙t)U˙(t)+e2tψ(t)eU(t)1−eU(t)=0.(3.3.19)t→+∞Similarly,from(3.3.13)and(3.3.16)weobtain:limψ(˙t)U˙(t)=0.t→−∞While(3.3.12)and(3.3.16)imply:
2tU(t)U(t)2(N+1)tψ(t)eψ(t)e1−e=Ote,ast→−∞.tThus,

limψ(˙t)U˙(t)+e2tψ(t)eU(t)1−eU(t)=0,(3.3.20)t→−∞andtogetherwith(3.3.18),(3.3.19),and(3.3.20)weconclude!+∞
e2tψ(t)eU(t)1−eU(t)dt=0,−∞incontradictionwiththefactthat,e2tψ(t)eU(t)(1−eU(t))>0inR.Remark3.3.13Thenon-degeneracyresultofTheorem3.3.12waspointedoutin[CFL]andalsoin[T7].Next,weshowhowtouseTheorem3.3.12inordertoestablishauniquenessandnon-degeneracyresultfortopologicalsolutionof(3.1.4).Forthispurpose,recallthatthetopologicalboundaryconditions(3.1.9)canbeequivalentlyensuredbyconsideringsolutionsof(3.1.4)inL1(R2)(seeProposition3.2.4andRemark3.2.5).ForR>0set,B1=u∈L1(R2):||u||≤RRL1(R2)theballofradiusRinL1(R2).Wehave: 3.3Auniquenessresult103Theorem3.3.14ForeveryR>0thereexistλR>0andµR>0suchthatforλ≥λRanysolutionuof(3.1.4)inB1satisfies:R!||∇ϕ||2+λeu(2eu−1)ϕ2≥µ2122R||ϕ||H1(R2),∀ϕ∈H(R).R2Inparticularlettingu=u0+v,thenvdefinesastrictlocalminimumforIλinH1(R2).ObservethatTheorem3.3.14appliesinparticulartothemaximalsolutionuλinCorollary3.2.8.Itturnsoutthatuλistheonlysolutionof(3.1.4)withboundedL1(R2)-normuniformlywithrespecttoλ.Infact,fromTheorem3.3.14wededucethefollowing:Theorem3.3.15Ifu˜λisasolutionfor(3.1.4),(3.1.9)andu˜λ=uλ,then||˜uλ||L1(R2)→+∞,asλ→+∞.(3.3.21)ProofofTheorem3.3.15.Arguebycontradiction,andsupposethatbesidesthemaxi-malsolutionuλ=u0+vλinCorollary3.2.8thereexistsasecondsolutionu˜λ=u0+˜vλof(3.1.4)inB1forλ=λn→+∞andR>0sufficientlylarge.ObservethatRvλ(z)0issufficientlylarge.However,thequestionofuniquenessforproblem(3.1.4),(3.1.9)foranyfixedλ>0remainsopen.ProofofTheorem3.3.14.Wearguebycontradictionandsupposethereexistsequences{λn}⊂(1,+∞)and{un}⊂L1(R2)satisfyingN−uunun2n=λne(1−e)−4πδzinR,(3.3.22)jj=1inthesenseofdistributions,limλn=+∞(3.3.23)n→+∞lim||un||L1(R2)<+∞(3.3.24)n→+∞suchthatsetting,(!)||∇ϕ||2+λunun212(3.3.25)µn=inf2ne(2e−1)ϕ∀ϕ∈H(R)||ϕ||H1(R2)=1R2 1043PlanarSelfdualChern–SimonsVorticeswehave−∞≤limµn≤0.(3.3.26)n→∞Wecanusetheanalysisoftheprevioussectionforuntoseeinparticularthat!!u2ununun2n<0inRandλne(1−e)+λn(1−e)≤c0,(3.3.27)R2R2withc0>0asuitableconstant(dependingonlyonthemultiplicityandthetotalnumberofthevortexpoints),and+max{|∇uun||un||L1(R2)n|+|un|+λn(1−e)}→0asn→∞,(3.3.28)δforδ=R2∪NBδ(zj)andδ>0.j=1Claim1:Forasuitableconstantc>0and∀n∈Nwehave,µn≥−c.(3.3.29)Toestablish(3.3.29),takeϕ∈H1(R2)with||ϕ||H1(R2)=1,andbymeansof(3.3.27)estimate:!eun(2eun−1)ϕ2R2!!!=2eun(eun−1)ϕ2+(eun−1)ϕ2+ϕ2R2R2R2!1!1!1!222≥−2e2un(eun−1)2+(eun−1)2ϕ4+ϕ2R2R2R2R2!1!1!1!222≥−2eun(1−eun)+(1−eun)2ϕ4+ϕ2R2R2R2R2!1!c02≥−3ϕ4+ϕ2.λnR2R2BytheSobolevinequality
!142ϕ≤2||ϕ||L2(R2)||∇ϕ||L2(R2)≤2||ϕ||L2(R2),R2wededuce!2+λunun2||∇ϕ||L2(R2)ne(2e−1)ϕ(3.3.30)R2!!11√2≥1+1−ϕ2c2λn−60λnϕλnR2R29c0≥1−≥−c,1−1λnforsuitablec>0,andsufficientlylargen.Therefore,(3.3.29)holds. 3.3Auniquenessresult105Asaconsequenceof(3.3.29),(3.3.26)weset:µ0:=limµn≤0.(3.3.31)n→∞Furthermorebythegivenpropertiesofun,weseethattheextremalproblem(3.3.25)attainsitsinfimumatafunctionϕn∈H1(R2)satisfying:⎧⎨λnununµn2− ϕn=e(1−2e)ϕn+ϕninR1−µn1−µn(3.3.32)⎩2ϕn>0inR,||ϕn||H1(R2)=1.Using(3.3.30)withϕ=ϕnwefind:'
'2+λunun212µn=||∇ϕn||L2nR2e(2e−1)ϕn≥1−1−λnλnR2ϕn√'1−6c0λn2ϕn22.RSo,bymeansof(3.3.31)wededucethat!λ2nϕn≤A,(3.3.33)R2∀n∈NandsuitableA>0.Claim2:Thereexistj0∈{1,...,N}andr0>0suchthat,∀ρ∈(0,r0],wefindaconstantaρ>0:!ϕ2≥aλnnρ.(3.3.34)Bρ(zj0)Toobtain(3.3.34),wearguebycontradictionandsupposethereexistsδ>0suffi-cientlysmallsuchthat!λ2nϕn→0,asn→∞.(3.3.35)∪NBδ(zj)j=1Asaconsequenceof(3.3.35)and(3.3.28)wehave:!!unun22λne(2e−1)ϕn−λnϕnR2!R2!unun2un2=2λne(e−1)ϕn+λn(e−1)ϕn!R2!R2!eun(1−eun)ϕ2+λun22≤2λnnn(1−e)ϕn+3λnϕnδδ∪NBδ(zj)j=1!
≤3supλun22n(1−e)||ϕn||L2+λnϕnδ∪NBδ(zj)j=1!λun2≤3supn(1−e)+λnϕn→0asn→∞.δ∪NBδ(zj)j=1 1063PlanarSelfdualChern–SimonsVorticesTherefore,asn→∞,!µ22n=||∇ϕn||L2(R2)+λnϕn+o(1)≥1+o(1),R2incontradictionwith(3.3.31),and(3.3.35)isestablished.Byreplacingun(z)withun(z+zj),wecanalwayssupposethatzj=0and00denotebyν∈Nthecorrespondingmultiplicity.Furthermore,bytakingr0>0smallerifnecessary,wecanassumethattheoriginistheonlyvortexpointinBr(0).Thus,un0satisfies−uununn=λne(1−e)−4πνδz=0inBr(3.3.36)0!λunn(1−e)≤c0andun<0inBr,(3.3.37)0Br0forsuitablec0>0.Inaddition,∀ρ∈(0,r0)andfrom(3.3.28),weknowthatmax|uunn|+λn(1−e)→0(3.3.38)ρ≤|z|≤r0!λ(1−eun)2=4πν2+o(1),(3.3.39)nBr(0)0asn→∞.Wecarryoutablow-upanalysis,andconsiderthescaledfunction:zuˆn(z)=un√z∈Dn:=B√λ(3.3.40)nr0λnzϕˆn(z)=ϕn√z∈Dn.(3.3.41)λnWedecomposeuˆn(z)=2νlog|z|+ˆvn(z),z∈Dn,(3.3.42)withvˆnsatisfying−vˆ2νvˆn2νvˆnn=|z|e(1−|z|e)inDn(3.3.43)!(1−|z|2νevˆn)≤c0.(3.3.44)DnClaim3:Thefollowingholds:(a)vˆnisuniformlyboundedinC2,α-topology;(3.3.45)(b)∀δ>0,thereexistscδ>0:sup|ˆun|≤cδ;(3.3.46)DnBδ(0)(c)∀p≥1,thereexistsCp>0:||ˆun||Lp(R2)≤Cp.(3.3.47) 3.3Auniquenessresult107Toestablish(a),westartbyshowingthat∀R>0thereexistsCR>0suchthatinfvˆn≥−CR.(3.3.48)BRTothispurpose,supposebycontradictionthatforanyR>0sufficientlylarge,infvˆn→−∞,BRasn→+∞(possiblyalongasubsequence).Setfn=euˆn(1−euˆn).Then0≤fn<1inR2and,−vˆn=fninB2Rvˆ1n|∂B≤2νlog.2R2RHence,wecanusetheHarnackinequality(seeProposition5.2.8)toobtainconstantsγ∈(0,1)andC>0(independentofn)suchthatsupvˆn≤γinfvˆn+C.BRBR'Therefore,supBvˆn→−∞and(1−|x|2νevˆn)=πR2+o(1)asn→∞.ButRBRforRsufficientlylargethiscontradicts(3.3.44),and(3.3.48)follows.HencevˆnaswellasvˆnareuniformlyboundedinL2(R2),andwecanusewell-knownellipticloc2,αestimatestogetherwithabootstrapargumenttoobtainauniformboundforvˆninC-locnormasclaimed.Toestablish(b),againbycontradiction,supposethereexistszn∈DnBδ(0):uˆn(zn)→−∞.Hencevˆn(zn)→−∞,andby(3.3.48)weseethatnecessarily|zn|→+∞,asn→+∞.ThereforeforanyR>0,B2R(zn)⊂DnBδ(0),providedthatnissufficientlylarge.Asabove,wecanusetheHarnackinequalityforu˜n(z)=ˆun(zn+z),sinceitsatisfies:−u˜n=fn(zn+x)inB2R(0)u˜n|∂B<0,u˜n(0)→−∞,asn→∞.2RInthisway,wededucethatsupBu˜n→−∞andR!!(1−euˆn)=(1−eu˜n)=πR2+o(1)asn→∞,BR(zn)BR(0)foreveryR>0,incontradictionwith(3.3.44).Finally,combining(a)and(b),weseethat∀p≥1,||ˆun||Lp(BR)≤Cp,R 1083PlanarSelfdualChern–SimonsVorticesforasuitableconstantCp,R>0,dependingonlyonpandR.Whileforeveryδ>0,weestimate!!|ˆuuˆnn|≤(1+sup|ˆun|)(1−e)≤Cδ,{|z|≥δ}DnBδ(0)R2forsuitableCδ>0.Bytheaboveestimates(c)easilyfollows.Byvirtueof(3.3.45),wecanuseadiagonalizationprocesstoobtainasubsequenceofvˆn(denotedinthesameway)suchthatvˆ2;(3.3.49)n→vuniformlyinClocandforu(z)=2νlog|z|+v(z),wehave⎧⎪⎨−u=eu(1−eu)−4πνδz=0,inR2u<0,u∈Lp(R2),p≥1⎪⎩'2(1−eu)<+∞.RInotherwords,ucoincideswiththeuniqueradiallysymmetricsolutionofproblem(3.3.1),forwhichproperties(3.3.2),(3.3.3),and(3.3.4)holdaswellasTheorem3.3.12.Claim4:||ˆun−u||L2(Dn)+sup|ˆun−u|→0,asn→+∞.(3.3.50)DnToestablish(3.3.50),weobservethatuˆn−u=ˆvn−v,andso||ˆun−u||C2(BR)→0asn→+∞,(3.3.51)foreveryR>0.Ontheotherhand,sinceuˆn−uisuniformlyboundedinDnBδ(0)(see(3.3.46)),from(3.2.24)wededucetheestimate!!|ˆu22−2uuˆnu2n−u|≤(1+Cδ)||e||L∞(R2Bδ(0))(e−e),(3.3.52)DnBδ(0)DnwithsuitableCδ>0.But,!!!!(euˆn−eu)2=(euˆn−1)2+(eu−1)2−2(1−euˆn)(1−eu)DnDnDnDn!=4πν2+4πν2−2(1−euˆn)(1−eu)+o(1)→0asn→+∞,Dn 3.3Auniquenessresult109''since,bydominatedconvergence,wehave:(1−euˆn)(1−eu)→2(1−eu)2=DnR4πν2,asn→∞.Hence,||euˆn−eu||→0asn→+∞,L2(Dn)andby(3.3.51)and(3.3.52),weconcludethat||ˆun−u||L2(Dn)→0,asn→∞.Furthermore,||euˆn(1−euˆn)−eu(1−eu)||≤2||euˆn−eu||→0,asn→+∞.L2(Dn)L2(Dn)Thus,wehaveestablishedthat||ˆun−u||L2(Dn)+|| (uˆn−u)||L2(Dn)→0,asn→+∞.Therefore,bywell-knownellipticestimates,foreveryρ∈(0,r0)andz∈Bρ√λ,wenseethatB2(z)⊂Dn(fornlarge)andsup|ˆun−u|≤C||ˆun−u||L2(B2(z))+|| (uˆn−u)||L2(B2(z))B1(z)≤C||ˆun−u||L2(Dn)+|| (uˆn−u)||L2(Dn)withasuitableconstantC>0independentofzandn.Consequently,∀ρ∈(0,r0),wefind:sup|z|≤√λ|ˆun−u|→0,asn→+∞.Also,using(3.3.28)and(3.3.3),wenρseethatsup|ˆun−u|≤sup|ˆun|+sup|u|√√√√√λnρ≤|z|≤r0λnλnρ≤|z|≤r0λn|z|≥λnρ√|u−(1−ε)ρλn≤supn|+Cεe→0ρ≤|z|≤r0asn→∞,andso(3.3.50)isestablished.Observethat,asaconsequenceof(3.3.51),wealsohave:sup|euˆn(1−euˆn)−eu(1−eu)|+sup|euˆn−eu|→0,asn→+∞.(3.3.53)DnDn
Next,weturntoanalyzetheasymptoticbehaviorofϕˆn=ϕn√zwhichweknowλntosatisfy:1euˆn(1−2euˆn)ϕˆµn−ϕˆn=n+ϕˆninDn1−µnλn(1−µn)(3.3.54)ϕˆn>0.Recalling(3.3.33),wederivethefollowinguniformestimate,!||∇ˆϕ222n||L2(Dn)+||ˆϕn||L2(Dn)=||∇ϕn||L2(Br)+λnϕn≤C0Br0forsuitableC>0. 1103PlanarSelfdualChern–SimonsVorticesHence,usingoncemoreellipticestimates,weseethatϕˆnisuniformlybounded2,αinC-topologyand,bypassingtoasubsequenceifnecessary(denotedinthesamelocway),wecanassumethatϕˆ2.n→ϕinClocFurthermore,ϕsatisfies:− ϕ=1eu(1−2eu)ϕinR21+|µ0|ϕ∈H1(R2),ϕ≥0.Byvirtueof(3.3.6),weseethatµ0=0,andsowecanuseTheorem3.3.12toconcludethatϕ=0.Ontheotherhand,usingthecut-offfunctionχ∈C∞(R2)suchthatχ=1inB1,0χ=0inR2Br0wedefine:2and0≤χ≤1,forρ∈0,2zχn(z)=χ√.ρλnThenwefind,!!!1uˆnuˆn2µn2(−ϕˆn)(χnϕˆn)=e(1−2e)χnϕˆn+χnϕˆnDn1−µnDnλn(1−µn)Dn!!2uˆnuˆn21uˆn2=e(1−e)χnϕˆn+(1−e)χnϕˆn1−µnDn1−µnDn!12−χnϕˆn+o(1)1−µnDn!!!=2eu(1−eu)χnϕˆn2+(1−eu)χnϕˆn2−χnϕˆn2+o(1).DnDnDnBytheexponentialdecayestimatein(3.3.3),weget!!2euuuχ22−(1−ε)R(1−e)+(1−e)nϕˆn≤3ϕˆn+Cεe,DnBRforeveryR>0andwithCεdependingonε∈(0,1)only.Sinceϕˆn2→0uniformlyinC2,fromtheestimateabovewededucethatloc!2eu(1−eu)+(1−eu)χ2nϕˆn→0,asn→∞.DnOntheotherhand,observingthat!!!212( ϕˆn)χnϕˆn=χn|∇ˆϕn|−∇χn·∇ˆϕnDnDn2Dn!!212=χn|∇ˆϕn|+ χn·ˆϕnDn2Dn!!2|| χ||L∞2C≥χn|∇ˆϕn|−2ϕˆn≥−Dn2ρλnBrλn0 3.4Planarnon-topologicalChern–Simonsvortices111withasuitableconstantC>0,wecancombinetheestimatesabovetodeducethat'χnϕˆn2→0,asn→+∞.DnConsequently,!!!ϕ2=ϕˆ2≤χ2λnn√nnϕˆn→0,asn→+∞,Bρ{|z|≤ρλn}Dnandrecalling(3.3.34),wearriveatacontradictionandconcludetheproofofTheorem3.3.14.WementionthatTheorem3.3.14andTheorem3.3.15aretheresultsofideasre-centlyintroducedbytheauthorin[T7]toestablishuniquenessoftopological-typesolutionsintheperiodiccase.3.4Planarnon-topologicalChern–SimonsvorticesInthissectionwetreat“non-topological”Chern–Simonsvortices,namely,solutionsto(3.1.4),(3.1.5),and(3.1.8)subjecttotheboundarycondition(3.1.10).WepresenttheperturbativeapproachofChae–Imanuvilov[ChI1],successfullyusedtohandleotherChern–Simonsmodels(cf.[ChI2],[ChI3],and[Ch3]),electroweakvorticesandstrings(cf.[ChT1]and[ChT2]),aswellascosmicstrings(see[Ch1],[Ch4],and[ChCh1]).Seealso[Ch2]foranapplicationofsuchanapproachinthecontextoftheBorn–Infieldtheory.Tobemorespecific,wefocusasabove(withoutlossofgenerality)tothecaseν=1.TheapproachintroducedbyChae–Imanuvilovin[ChI1]toobtain“non-topological”Chern–Simonsvorticesisbasedontheobservationthatifwehaveasolutionufor(3.1.4),thenforeveryε>0,thescaledfunction
z1uε(z)=u+2logεεsatisfiesNuε−λε2e2uε−4πδ2−uε=λeεzinR.(3.4.1)jj=1Therefore,wemayregardtheε-scaledproblem(3.4.1)asaperturbationofthe“sin-gular”Liouvilleproblem:⎧⎪⎪N⎨−u=λeu−4πδεzj(3.4.2)⎪⎪'j=1⎩uR2e<∞. 1123PlanarSelfdualChern–SimonsVorticesThisobservationalreadyputsanemphasisontheroleplayedbyLiouvilleequationsinthesearchfornon-topologicalsolutions.Thisfactwillbefurtherexploitedfortheperiodiccaseandintheelectroweakmodel.ObservethatbyLiouvilleformula(2.2.3),wecanexhibitanexplicitsolutionfor(3.4.2).Indeed,let$N!zf(z)=(N+1)(z−zj)andF(z)=f(ξ)dξ,0j=1andset$N!zfε(z)=(N+1)(z−εzj)andFε(z)=fε(ξ)dξ.0j=1By(2.2.3)weknowthat08|fε(z)|2uε,a(z)=logλ(1+|F22(3.4.3)ε(z)+a|)satisfies(3.4.2)foranyε∈Randa∈C.Incidentally,noticethatallsolutionsof(3.4.2)areobtainedinthisway(cf.[PT]andsee[CW],[CL1],[CL2],and[CK1]).Thus,wecanreasonablysearchforthesolutionofourproblem⎧⎪⎪N⎪⎪−u=λeu(1−eu)−4πδεz⎨jj=1(3.4.4)⎪⎪⎪⎪u(z)→−∞as|z|→+∞⎩uu12e(1−e)∈L(R)intheformu(z)=u0(εz)+logε2+ε2w(εz),(3.4.5)ε,awithwasuitablefunctionthatidentifiestheerrortermandsatisfieseε2w−1u0ε,a2(u0ε,a+ε2w)− w=λe−λe.(3.4.6)ε2Weconsiderthefreeparametersεandaaspartofourunknownsandconcentratearoundthevaluesε=0anda=0where(3.4.6)reducesto12 w+ρw=ρ(3.4.7)λu0withρ=λeε=0,a=0,theradialfunction,givenasfollows:8(N+1)2r2Nρ(r)=2,r=|z|.(3.4.8)1+r2(N+1) 3.4Planarnon-topologicalChern–Simonsvortices113Remark3.4.17Forlateruse,noticethatthedefinitionofρ,aswellas(3.4.7),makegoodsensealsoforN=0.Wecananalyze(3.4.7)withintheclassofradialfunctions,wherewefindanexplicitsolutionw0=w0(r)givenbytheexpression(3.4.30)below.Thus,usingthedecompositionw(z)=w0(|z|)+u1(z),(3.4.9)weneedtosolveforu1inthefollowingequation:eε2w0+ε2u1−1u0P(u1,a,ε):=u1+λeε,a−ρw0ε2+λe2(u0ε,a+ε2w0+ε2u1)−1ρ2=0.(3.4.10)λToattainthisgoal,weaimtoapplytheImplicitFunctionTheorem(cf.[Nir])totheoperatorPconsideredonsuitablefunctionalspaceswhereitextendssmoothlyatε=0tosatisfyP(0,0,0)=0.Tothispurpose,Chae–Imanuvilovin[ChI1]haveintroducedthespacesXα=u∈L2(R2):(1+|z|2+α)u2∈L1(R2),α>0loc()(3.4.11)Y2,2(R2u22α=u∈W):u∈Xα,α∈L(R),α>0loc(1+|z|)1+2equippedrespectivelywiththescalarproduct!!2+αuv(u,v)X=(1+|z|)uvand(u,v)Y=( u, v)X+,ααα2+αR2R2(1+|z|)andrelativenormsdenotedby||·||Xand||·||Y,respectively.ααForanyα>0,thefollowingcontinuousembeddingpropertieshold:Xα→Lq(R2),∀q∈[1,2);Yα→C0(R2).locFurthermore,wehave:Lemma3.4.18Letα∈(0,1)andv∈Yα.(a)Ifvisharmonic,thenvisaconstant.(b)Thefollowingestimateshold:|v(z)|≤C||v||log(1+|z|),inR2;(3.4.12)Yα||∇v||Lp≤Cp||v||Y,foreveryp>2.(3.4.13)αwhereC>0andCp>0aresuitableconstantsdependingonαand(α,p)respectively. 1143PlanarSelfdualChern–SimonsVorticesProof.Toestablish(a),wefirstobservethatbystandardellipticregularity,anyhar-monicfunctionvinYαissmooth.Moreover,ifweexpressvaccordingtoitsFourierdecomposition+∞ξiθkiθv(z)=k(r)e,z=re,k=−∞withξk=ξk(r)∈Csuchthatξ−k=ξkandsatisfies1k2φ¨+φ˙−φ=0,∀k∈Z+.(3.4.14)rr2Notethatfork≥1,thefunctionsφk11,k=randφ2,k=krepresentafundamentalrsetofsolutionsto(3.4.14),whilefork=0wehaveφ1,0=1andφ2,0(r)=log(r)asfundamentalsolutions.ThesmoothessofvinR2andthefactthatv∈L2(R2)for1+r1+αα∈(0,1)implythatξk≡0foreveryk∈N,andξ0=constant,sothat(a)follows.Toobtain(b),letv∈Yαsothat v:=g∈Xα.Set!1v(˜z)=log|z−η|g(η)dη,(3.4.15)2πR2thenv˜= vinR2.Weestimate:!12+α1|log|z−η|||˜v(z)|≤(1+|η|)2|g(η)|12πR2(1+|η|2+α)2!1!21212+α22log|z−η|≤(1+|η|)g(η)2+α2πR2R2(1+|η|)!1221log|z−η|≤||v||Y.(3.4.16)α2+α2πR2(1+|η|)Inturn,for|z|>1theintegralabovecanbeestimatedasfollows:!2log|z−η|2+αR2(1+|η|)!!221log|z−η|≤log+{|z−η|<1}|z−η|{1<|z−η|≤2|z|}(1+|η|2+α)!2log|z−η|+{|z−η|>2|z|}(1+|η|2+α)!!111≤2πlog2rdr+log2(2|z|)2+α0rR21+|η|

!log2|η|1+|z||η|+{|η|≥|z|}(1+|η|2+α)!!221log(2|η|)≤log|z|++C.(3.4.17)2+α2+αR21+|η|R2(1+|η|) 3.4Planarnon-topologicalChern–Simonsvortices115Thus,bycombining(3.4.16)with(3.4.17)weconclude:|˜v(z)|≤Cα||v||Ylog(|z|+1).(3.4.18)αBydifferentiating(3.4.15)withrespecttozandusingwell-knownpotentialestimates(cf.[GT]),wegetthat,foreveryq∈(1,2),||∇˜v||2q≤C||v˜||Lq(R2)≤C|| v||Xα≤C||v||Yα(3.4.19)L2−q(R2)forasuitableconstantC>0dependingonαandq.Inparticular,from(3.4.18)itfollowsthat,v˜∈Yαand||˜v||Y≤Cα||v||YforasuitableconstantCα>0.Sov−˜vααdefinesaharmonicfunctioninYα,andforα∈(0,1),wecanusepart(a)toconcludethatc=v−˜v(3.4.20)forasuitableconstantc∈R,with||c||Y≤||v||Y+||˜v||Y≤Cα||v||Y.(3.4.21)ααααHence,itsufficestocombine(3.4.18),(3.4.19),(3.4.20),and(3.4.21)toobtainthedesiredconclusion.WorkingwiththespacesXαandYαisparticularlyadvantageousforthelinearoperatorL=+ρ:Yα−→Xα,(3.4.22)aswecancharacterizeexplicitlyKerL⊂YαandImL⊂Xα.Tothispurpose,considerthefamilyoffunctions:U0(µz)+logµ2,µ>0,anda∈C;µ,a(z)=uε=0,asatisfying:−U=λeU−4πNδz=0.(3.4.23)Lettingu0=Uµ=1,a=0=logρandusingpolarcoordinates,weseethatthefollowingfunctionsbelongtoKerLinYα,∀α>0:1∂1−r2(N+1)φ0=Uµ,a|µ=1,a=0=;2(N+1)∂µ1+r2(N+1)1∂rN+1cos((N+1)θ)φ+=−Uµ,a|µ=1,a=0=,(x=Rea);(3.4.24)4∂x1+r2(N+1)1∂rN+1sin((N+1)θ)φ−=−Uµ,a|µ=1,a=0=,(y=Ima).4∂y1+r2(N+1)Moreinterestingly,thefollowingholds. 1163PlanarSelfdualChern–SimonsVorticesProposition3.4.19Forα∈(0,1),theoperatorLin(3.4.22)satisfies:(a)KerL=span{φ0,φ+,φ−}⊂Yα;'(b)ImL={f∈Xα:R2fφ±=0}.ToderiveProposition3.4.19,westartbydescribingthebehavioroftheoperatorLoverradialfunctions.Hence,denotebyLr:Yαr→Xαrtheoperatord21dLrφ=φ+φ+ρφ,φ∈Yr,(3.4.25)dr2rdrαwhereYαrandXαrdenotethesubspacesinYαandXαrespectivelyrestrictedtoradialfunctions.Lemma3.4.20Letα∈(0,1)andn∈Z+.Then:(a)φ∈YαrsatisfiesLrφ=0ifandonlyifφ∈span{φ0}.(b)Lr:Yαr→Xαrisonto.Moreprecisely,forf∈Xαr,let!21rw(r)=φ0(r)logr+φ0(t)f(t)tdtN+11+r2(N+1)0!r21−φ0(r)φ0(t)logt+f(t)tdt.(3.4.26)0N+11+t2(N+1)Thenw∈Yαrandsatisfies:Lrw=f.Observethatw(r)andw(˙r)extendwithcontinuityatr=0wherewefind:w(0)=0=˙w(0).Furthermore,setting!+∞cf=φ0(t)f(t)tdt,(3.4.27)0wehave:Corollary3.4.21Thefunctionwin(3.4.26)admitsthefollowingasymptoticbehav-ior:w(r)=−cflogr+O(1),asr→+∞(3.4.28)cfw(˙r)=−+O(1),asr→+∞.(3.4.29)rInparticular,bytakingf(r)=1ρ2in(3.4.26)weobtainλ!12rw0(r)=(1−r2(N+1))logr+φ0(t)tρ2(t)dtλ1+r2(N+1)N+10-
!r2(N+1)22−1−rφ0(t)logt+ρ(t)tdt.0(N+1)1+t2(N+1)(3.4.30) 3.4Planarnon-topologicalChern–Simonsvortices117Thefunctionin(3.4.30)definesasolutionforproblem(3.4.7)inYαr,suchthat:c0w0(r)=−logr+O(1),asr→+∞;(3.4.31)λ1c0w˙0(r)=−+O(1),asr→+∞,(3.4.32)λrwith!+∞2N3sc0=16(N+1)ds>0.(3.4.33)0(1+sN+1)4Indeed,accordingtoCorollary3.4.21,weseethat!+∞c0=φ0(t)ρ2(t)tdt0

2!+∞1−t2(N+1)=8(N+1)25t4N+1dt01+t2(N+1)!+∞1−sN+1=32(N+1)4s2Nds0(1+sN+1)5!!+∞s2N+∞sN=32(N+1)4−2s2N+10(1+sN+1)40(1+sN+1)5!2N!4+∞s1+∞d12N+1=32(N+1)+sds0(1+sN+1)42(N+1)0ds(1+sN+1)4!!4+∞s2N2N+1+∞s2N=32(N+1)ds−0(1+sN+1)42(N+1)0(1+sN+1)4!+∞s2N=16(N+1)3ds,0(1+sN+1)4and(3.4.33)isestablished.Fromnowonweshallsubstitutesuchasolutionw0intothedefinitionoftheoper-atorPgivenin(3.4.10).ProofofLemma3.4.20.Toobtain(a),noticethatifφ∈Yαrsatisfies1φ¨+φ˙+ρφ=0,rthenbystandard(elliptic)regularitytheory,weknowthatφextendswithcontinuityatr=0.Consequentlyrφ˙∈C1[0,+∞),andweobtainφ∈C2(0,+∞)∩C1[0,+∞),andφ(˙0)=0.Nowwriteφ(r)=φ0(r)ψ(r), 1183PlanarSelfdualChern–SimonsVorticessothatψ∈C2((0,1)∪(1,+∞))∩C1([0,1)∪(1,+∞))satisfies:
ψ¨+ψ˙1φ˙0+2=0rφ0(3.4.34)ψ(˙0)=0.Consequentlydrφ2(r)ψ(˙r)=0,thatgivesdr0ψ(r)=Cfor0≤r<1;whileforr>1,wefindAψ(˙r)=,forasuitableconstantA∈R.(3.4.35)rφ2(r)0Thatis2ψ(r)=Alogr++B,forr>1(3.4.36)(N+1)1−r2(N+1)forsuitableconstantsA,B,C∈R.Inotherwords:Cφ0(r),0≤r<1φ(r)=2AAφ0(r)logr+(N+1)(1+r2(N+1))+Bφ0(r),r>1.Ontheotherhand,thecontinuityofφatr=1requiresthatA=0,whilethecontinu-ityofφ˙atr=1impliesthatB=C,andsoφ∈span{φ0}asclaimed.Todemonstratethattheformula(3.4.26)givesasolutionwinYαrforthenon-homogeneousequationLrw=f,weproceedasaboveandsetw=φ0ψ.ThereforeLrw=f,ifandonlyif,
d2rφ0(r)ψ˙=φ0(r)f(r)r.drRecalling(3.4.35)and(3.4.36),wemayintegratetheequationabovefor00.Andsincew(r)admitslogarithmicgrowthasr→+∞andf∈Xα,wecanensurethatitbelongstoYα.ProofofProposition3.4.19.Westarttoestablish(a).Lettingv∈YαsuchthatLv=0,thenbystandardellipticregularitytheoryweseethatv∈C2(R2).Wewritevaccord-ingtoitsFourierdecompositionvikθiθv(z)=k(r)e,z=re,(3.4.37)k∈Zwithcomplexvaluedfunctionsvk=vk(r)suchthatv−k=vkandwhoserealandimaginarypartsatisfyk2Lr(φ)−φ=0.(3.4.38)r2Ifk=0,thenfortherealvaluedradialfunctionv0(r)∈Yαr,wecanuseLemma3.4.20toseethatactuallyv0(r)∈span{φ0}.Fork∈N,wearegoingtodetermineafundamentalsetofsolutionsto(3.4.38)byusingafamilyofsolutionsforthe(singular)Liouvilleequations.Moreprecisely,fora∈Candk∈N,let8|(N+1)zN+(k+N+1)azN+k|2ψa,k(z)=log2,1+|zN+1+azN+k+1|2sothatψa=0,k=logρ.Accordingto(2.2.3),ψa,ksatisfiesk− ψa,k=eψa,k−4πNδz=0−4πδa,inR2,zjj=1whereza,j=1,...,kcorrespondtothek-distinctnon-zerorootsofthepolynomialj(N+1)zN+(k+N+1)azN+k.Noticeinparticularthat|za|→+∞asa→0,j∀j=1,...,k.Therefore,foreachtestfunctionϕ∈C∞(R2),and|a|sufficiently0small,wehave:− ψψa,kϕ−4πNϕ(0).a,kϕ=eSowecandifferentiatethisexpressionata=0withrespecttox=Reaandy=Ima,andobtainthat∂ψa,k2∂ψ2ϕ1,k=a=0=φk(r)coskθ;ϕ2,k==φk(r)sinkθ,∂xN+1∂ya=0N+1 1203PlanarSelfdualChern–SimonsVorticeswithk+N+1+(k−N−1)r2(N+1)kφk(r)=r,(3.4.39)1+r2(N+1)satisfyLϕ1,k=0=Lϕ2,k,∀k∈N.Asaconsequence,weobtainthatφk(r)satisfies(3.4.38).Inaddition,byreplacingkwith
−kin(3.4.39)westillobtainasolutionφ˜kfor(3.4.38)andwecheckthatφ˜k(r)=1.Thus,φφkk(r)andφ˜k(r)defineafundamentalsetofsolutionsfor(3.4.38).Onrtheotherhand,therealandimaginarypartofvkcannotincludethecomponentφ˜k(r),whichadmitsa1singularityattheorigin,∀k∈N.Furthermore,forα∈(0,1)rkandk=N+1,thefunctionφk∈/Yαrsinceitbehavesasrk,asr→+∞.Soweconcludethatvk=0,∀k=N+1.Finally,fork=N+1weseethatRe(vk=N+1)rN+1andIm(vk=N+1)belongtospan{φN+1(r)}=span1+r2(N+1).Weconcludethatv∈span{φ0,φ+,φ−}asclaimed.Toestablish(b),westartbyobservingthefollowingfact:Claim:TherangeofLisclosedinXα.Letϕn∈(KerL)⊥⊂YαbesuchthatLϕn=fn∈Xαandfn→finXα.WeclaimthatϕnisuniformlyboundedinYα.Thisisequivalenttosayingthat'ϕ2n≤C,∀n∈N,forsuitableC>0.Arguingbycontradiction,assumeR2(1+|z|α+2)that(alongasubsequence)!122ϕnϕncn=→+∞,andsetφn=.α+2R2(1+|z|)cnSothat,!2φn⊥=1,φn∈(KerL)⊂Yα,andLφn→0inXα.(3.4.40)α+2R2(1+|z|)Asaconsequenceof(3.4.40),weseethatφnisuniformlyboundedinYα.Hence,wefindφ∈Yαsuchthatalongasubsequence,wehaveφn→φweaklyinYα,andwecanassumefurtherthattheconvergenceabovealsoholdsinL2(R2).locSince|φn(z)−φ(z)|≤c(||φn||Y+||φ||Y)log(1+|z|)≤Clog(1+|z|),ααweseethat!2!(φn−φ)2log(1+|z|)21+α2≤||φn−φ||L2(BR)+C1+α2=||φn−φ||L2(BR)+o(1),R2(1+|z|)|z|≥R(1+|z|) 3.4Planarnon-topologicalChern–Simonsvortices121'2(φn−φ)asR→+∞.Thus,R2(1+|z|1+α)2→0asn→∞,andweconcludethat!2|φ|=1.(3.4.41)1+α2R2(1+|z|)''Similarlyweseethat,R2ρφn→R2ρφasn→∞.Hence,Lφ=0andφ∈(KerL)⊥,thatis,φ=0incontradictionto(3.4.41).Inconclusion,thesequenceϕn∈(KerL)⊥isuniformlyboundedinYα,andwecanargueexactlyasabovetofindϕ∈Yαsuchthatϕn→ϕweaklyinYαwithLϕ=f,andtheClaimisproved.Consequently,wemaydecomposeX⊥,α=ImL⊕(ImL)accordingtothescalarproductinXα.Thus,forξ∈(ImL)⊥wehave=(Lu,(1+|z|)2+αξ),∀u∈Y0=(Lu,ξ)Xα.αThedensityofC∞inYαimpliesthat02+αξ∈Yψ=(1+|z|)αandLψ=0.Thereforebypart(a),wemaywrite:ψ=a0φ0+a+φ++a−φ−forsuitablecon-stantsa0,a+anda−.Furthermore,byLemma3.4.20,weknowthattheradialfunction1∈ImLandsatisfies(f,φf(r)=φ0(r)(1+r2)2±)L2=0.Consequently,!2+∞φ0(r)0=(f,ξ)Xα=(f,ψ)L2=a02π2rdr,01+rthatis,a0=0,andpart(b)ofourstatementfollows.Atthispointwecancompleteourperturbationanalysis.Byvirtueof(3.4.30)and(3.4.33),weseethatP:Yα×C×R−→Xαin(3.4.10)isawell-definedsmoothoperatorthatcanbeextendedwithcontinuityatε=0,wherewehaveP(0,0,0)=0.Moreover,thelinearizedoperator∂PA:(0,0,0):Yα×C−→Xα(3.4.42)∂(u1,a)takestheformA(ϕ,b)=Lϕ+M(b),(3.4.43)withϕ∈Yα,b=b1+ib2∈Cand2222M(b)=−4ρw0−ρφ+b1−4ρw0−ρφ−b2.(3.4.44)λλSoonthebasisofProposition3.4.19,weexpectalsotocharacterizeKerA⊂YαandImA⊂Xα.Tothispurpose,weobserve: 1223PlanarSelfdualChern–SimonsVorticesLemma3.4.22!2!+∞2r2(N+1)ρwρ22=πρ2(r)rdr<0.0−φ±ρ(r)w0(r)−2R2λ0λ1+r2(N+1)Proof.Westartbyobservingthatr1(N+1)2r4N+2L2=4.161+r2(N+1)1+r2(N+1)Therefore,inviewofthedecayestimates(3.4.31)and(3.4.32)ofw0,wecanuseintegrationbypartstoobtain!222ρw0−ρφ±R2λ!+∞8(N+1)2r4N+222r2(N+1)=π4w0(r)−ρ2rdr01+r2(N+1)λ1+r2(N+1)!+∞112r2(N+1)=πLrwρ220(r)−2rdr021+r2(N+1)λ1+r2(N+1)!+∞112r2(N+1)rw2=L02−ρ2rdr021+r2(N+1)λ1+r2(N+1)!1+∞12r2(N+1)=ρ2(r)−rdr,22λ021+r2(N+1)1+r2(N+1)where,inthelastidentity,wehaveusedthefactthatw0satisfies(3.4.7).Toestimatethelastintegralabove,weusethechangeofvariablet=r2tofind!+∞1−4r2(N+1)ρ2(r)2rdr021+r2(N+1)!+∞1−4tN+1=16(N+1)4t2Ndt0(1+tN+1)6!!+∞t2N+∞t3N+1=16(N+1)4−5dt0(1+tN+1)50(1+tN+1)6!2N!4+∞t1+∞2N+1d1=16(N+1)+tdt0(1+tN+1)5N+10dt(1+tN+1)5!!4+∞t2N2N+1+∞t2N=16(N+1)−dt0(1+tN+1)5N+10(1+tN+1)5!+∞t2N=−16(N+1)3Ndt<0,0(1+tN+1)5andthedesiredconclusionfollows. 3.4Planarnon-topologicalChern–Simonsvortices123Wearenowreadytoconlcude:Proposition3.4.23Forα∈(0,1),theoperatorA:Yα→Xαin(3.4.42)and(3.4.43)isontoandKerA=KerL=span{φ0,φ+,φ−}.Proof.Letf∈Xα,weneedtofindϕ∈Yαandb=b1+ib2∈CsuchthatA(ϕ,b)=Lϕ+M(b)=f,withM(b)in(3.4.44).(3.4.45)Tothispurpose,multiply(3.4.45)byφ+andintegrateoverR2tofind!!!!22222fφ+=Lϕφ+−4b1ρw0−ρφ+−4b2ρw0−ρφ+φ−R2R2R2λR2λ!+∞2r2(N+1)=−4πb21ρ(r)w0(r)−ρ(r)dr,0λ(1+r2(N+1))2asfollowsbyProposition3.4.19(b),andthewell-knownorthogonalitypropertiesoftrigonometricfunctions.OnthebasisofLemma3.4.22,wemaysolveforb1andderive:!!−11+∞2r2(N+1)b21=−fφ+ρ(r)w0(r)−ρ(r)dr.(3.4.46)2(N+1)24πR20λ(1+r)Analogously,multiplying(3.4.45)byφ−andintegratingoverR2wederive:!!−11+∞2r2(N+1)b22=−fφ−ρ(r)w0(r)−ρ(r)2dr.(3.4.47)4πR20λ1+r2(N+1)Setg=f−M(b),withM(b)in(3.4.44)andb1andb2specified,respectively,'in(3.4.46)and(3.4.47).Weeasilycheckthat,R2gφ±=0.Ourproblemisnowreducedtofindingϕ∈Xα:Lϕ=g,andthiswehavebyProposition3.4.19(b).Thus,wehavededucedthatImA=Xα.TocharacterizeKerA,justtakef=0intheargumentabovetofindthatb1=0=b2,andsoAϕ=0,ifandonlyif,Lϕ=0;thatisKerA=KerL=span{φ0,φ+,φ−}.SettingU⊥α=span{φ0,φ+,φ−},weobtainthefollowingexistenceresultfor(3.4.4):Proposition3.4.24Foreveryλ>0andα∈(0,1),thereexistε0>0sufficientlysmallandsmoothfunctionsaε:(−ε0,ε0)→C,u1,ε:(−ε0,ε0)→Uα,withaε=0=0andu1,ε=0=0,suchthatthefunction0(εz)+logε2+ε2w2uε(z)=uε,aε0(ε|z|)+εu1,ε(εz)(3.4.48) 1243PlanarSelfdualChern–SimonsVorticesdefinesasolutionfor(3.4.4),withw0definedin(3.4.30)andsatisfying(3.4.31)-(3.4.33).Furthermoreasε→0,thefollowingestimateshold:|uo(1),∀z∈R2;1,ε(z)|≤o(1)log(1+|z|),|∇u1,ε(z)|=1+|z|(3.4.49)'λ2euε(1−euε)=8π(N+1)+o(1).RProof.AstraightforwordapplicationoftheImplicitFunctionTheorem(cf.[Nir])yieldstouεin(3.4.48).Soitremainstocheckthevalidityof(3.4.49).Tothispur-pose,from(3.4.12),wehave|ulog(1+|z|),∀z∈R2,1,ε(z)|≤C1||u1,ε||Yαand||u1,ε||Y→0,asε→0.Furthermore,from(3.4.10)weseethat,α−u1,ε=f1,ε,withf1,ε→0inXα.Wecancheckalsothat(1+|z|2)|f1,ε|≤cλεinR2,withasuitableconstantcλdependingonλbutindependentofε.Sinceu1,ε∈Yα,wecanwrite!1y−z∇u1,ε(z)=f1,ε(y)dy,22πR2|y−z|andfor|z|≥2derivethefollowingestimate:!!|z||f1,ε(y)||z||f1,ε(y)||z||∇u1,ε(z)|≤+{|y−z|≤|z|}|y−z|{|y−z|≥|z|}|y−z|22!|y||f1,ε(y)|≤2{|y−z|≤|z|}|y−z|2⎛⎞!+2|f1,ε(y)|+π|z|⎝max|y||f1,ε(y)|⎠+2||f1,ε||L1(R2){|y−z|≤|z|}|z|2{|y|≥2}≤cλε+2||f1,ε||L1(R2)→0,asε→0.Thefirstestimatein(3.4.49)thenfollows.Finally,noticethat!!uzz1λeε(z)(1−euε(z))=λeuε(ε)(1−euε(ε))2R2!R2ε!uε,aε(z)+ε2(w0+u1,ε)22u0ε,aε(z)+2ε2(w0+u1,ε)=λe−λεe.R2R2Therefore,wecanusetheavailableestimatesand(bydominatedconverge)passtothelimitintotheintegralsignandconclude:!!!+∞2N+1uε(z)uε(z)u0rlimλe(1−e)=λeε=0,a=0=16π(N+1)dr2ε→0R2R201+r2(N+1)!+∞dt=8π(N+1)=8π(N+1).0(1+t)2 3.4Planarnon-topologicalChern–Simonsvortices125ByvirtueofProposition3.4.24,weconcludethefollowingexistenceresultcon-cerningnon-topologicalChern–Simonsvortices:Theorem3.4.25Foragivensetof(vortex)pointsZ={z1,...,zN}(repeatedac-cordingtotheirmultiplicity)andk>0,thereexistε0>0suchthat,foreveryε∈(0,ε0),wehaveaplanarvortexconfiguration(Aε,φε)±solutiontotheself-dualequations(1.2.45)inR2(withthe±signchoosenaccordingly)suchthati)φ±εvanishesexactlyinthesetZ,andifnj∈Nisthemultiplicityofzj,thenεεnjφ+(z)andφ−(z)=O((z−zj)),asz→zj,j=1,...,N.(3.4.50)ii)φ±εsatisfiesproperty(3.2.1),andφ±ε(z)→0as|z|→+∞.Moreprecisely,thereexistconstantsCε>0,Rε>0andβε→0+asε→0suchthatε|+|z|2|∇|φε||2+|φε|2≤C−2(N+2+βε)|F12±±ε|z|,∀|z|≥Rε.(3.4.51)iii)Magneticflux:!ε=(Fε)±12±=±4π(N+1)+o(1);(3.4.52)R2Electriccharge:!Qε=(J0)±ε±=±4πk(N+1)+o(1);(3.4.53)R2Totalenergy:!E2ε=E±=4πν(N+1)+o(1);(3.4.54)R2asε→0.ObservethatincomparisontoTheorem3.2.3,theconstructionaboveyieldstoavortexconfigurationthatverifiesthe“concentration”property(3.2.8)onlywhenallthevortexpointscoincide.Indeed,tofixtheideas,letz1=z2=···=zN=0,ν=1andλ=1,thenbyLemma3.4.20wecanargueasabovetoobtainaradialsolution(abouttheorigin)fortheequationu(1−eu)−4πNδ2−u=ez=0,inR,(3.4.55)intheform8(N+1)2ε2(N+1)r2N22uε(r)=log+εw0(εr)+εu1,ε(εr),(3.4.56)(1+ε2(N+1)r2(N+1))2withw0definedin(3.4.30)andu1,εsatisfyingtheestimatesin(3.4.49).Consequently,fork>0,thefunction2uε,k(r)=uεr(3.4.57)k 1263PlanarSelfdualChern–SimonsVorticesdefinesa(radial)solutiontotheproblem:−u=4eu(1−eu)−4πNδ22z=0inRk(3.4.58)u(z)→0as|z|→+∞.εk→0,ask→0,andsetuThus,ifwechooseε=εk:kk=uεk,k,then4euk→8π(N+1)δ2+z=0,weaklyinthesenseofmeasureinR,ask→0;k2andthecorrespondingnon-topologicalChern–Simonsradialvortexconfigurationsat-isfies(3.2.8)withz1=z2=···=zN=0.Werecallthataclassofradial“non-topological”vorticeswasconstructedforthefirsttimebySpruckandYangin[SY1]byusinga“shooting”methodforthecorre-spondingO.D.E.problem.Inparticular,in[SY1]itwasshownthat(forν=1)theenergyEofradialnon-topologicalsolutionssatisfiesthelowerboundE>4π(N+1),whichwecannowasserttobesharp.Howevertohavenon-topologicalvorticesenhancedwithproperty(3.2.8),(asitwouldbedesirableforthephysicalapplications)seemstorequireabiggeramountofenergy,i.e.,E>8πN.ThisfactwasobservedbyChan–Fu–Linin[CFL],whointroducedanalternativeconstructionofnon-topologicalvorticesrelatedtoanewclassofradialsolutionsfor(3.4.55).Tothispurpose,weobservethattheradialsolutionuεin(3.4.56)satisfies:!+∞euε(r)(1−euε(r))rdr=4(N+1)+o(1)asε→0.0In[CFL]theauthorsshowthat,infact,foranyprescribedβ>4(N+1),thereexistsauniqueradialsolutionu=u(r)fortheequation(3.4.55)satisfying:!+∞eu(r)(1−eu(r))rdr=β0(seeTheorem2.1in[CFL]).Hence,Chan–Fu–Linin[CFL]searchforthesolutionof(3.1.4),whoseprofile(aftersuitablescalingandtranslation)aroundavortexpointlookslikeonesuch“new”radialsolution.However,toattainsuch“concentration”property,theauthorsneedtorequireaspecificlocationforthevortexpoints{z1,...,zN},whichmustbeplacedaroundacircletoformanequilateralpolygonwithZN-symmetry.Thus,ifz0isthecenterofthecircle,thentheanglebetweenthesegmentz2π,0zjandz0zj+1mustbeequaltoN∀j=1,...,NandzN+1=z1.ThesolutionconstructedinthiswaykeepsthesameZN-symmetry.Moreprecisely,undersuchassumptionsonthelocationofthevortexpointsandthenormalizationν=1,thefollowingholds: 3.5Finalremarksandopenproblems127Theorem3.4.26[CFL]:Givenanumber>8πN,thereexistsk0>0suchthat,foreveryk∈(0,k0),wehaveaplanarselfdualvortexconfiguration(A,φ)±solutionto(1.2.45)inR2(with±signchosenaccordinglyandν=1)withthefollowingproperties:i)φ±vanishesexactlyattheZN-symmetricallylocated(distinct)pointsz1,...,zNandφ+(z),φ−(z)=O(z−zj)asz→zj,j=1,...,N.Furthermore|φ±|<1inR2and|φ±|→0as|z|→+∞.ii)Thefollowingestimateholds:
−2−N2|∇|φ222π|(F12)±|+|z|±||+|φ±|=O|z|,as|z|→+∞.iii)Thecorrespondingmagneticflux,electricchargeandtotalenergyaregivenre-spectivelyasfollows:''Magneticflux:2(F12)±=±;Electriccharge:2(J0)±=±k;'RRTotalenergy:R2E±=.Infact,forthelocalfluxaroundzj,j=1,...,N,wehave!1+(F12)±=±+o(1),ask→0,Bδ(zj)N0forsufficientlysmallδ0>0.iv)Ask→0+,wehave:
1N(a)(F12)±→±δz,weaklyinthesenseofmeasure;Nj=1j(b)log|φ|2islocallyradiallysymmetricaroundzj,j=1,...,N,andaftersuitablescaling,N2→2−log|z−z2log|φ|j|+C,2πNj=1forsomeC∈R.Wereferto[CFL]fordetails.3.5FinalremarksandopenproblemsInconcludingourdiscussionaboutplanarChern–Simonsvortices,wewishtoreturnandemphasizethemainproblemswhichremainopenfortheplanar6th-orderChern–Simonsmodeldiscussedabove.Firstly,concerningplanartopologicalsolutions,theiruniqueandpossiblysmoothdependenceonthevortexpointsremainstobeclarified.Thisquestioncanbeformu-latedmorepreciselyintermsoftheellipticproblem 1283PlanarSelfdualChern–SimonsVortices⎧⎪⎪N⎨u(1−eu)−4πδ2−u=ezinRj(3.5.1)⎪⎪j=1⎩u(x)→0as|x|→+∞asfollows:Openproblem:Foranyassignedsetof(vortex)points{z1,...,zN}⊂R2(notnec-essarilydistinct),doesproblem(3.5.1)admitauniquesolution?Canitbesmoothlyparametrizedbythegivenpoints{z1,...,zN}?Weknowtheanswertobeaffirmativeforthecasez1=···=zN,namely,whenallthevortexpointsaresuperimposedwithmultiplicityN∈N.Infact,inthiscaseonecanprovideuniquenessforproblem(3.3.1)foranyN∈[0,+∞),andinparticular,forN=0,whereDiracmeasuresarenotincludedin(3.5.1).Suchuniquenesspropertypersistsundersmall“perturbations,”inthesensethat,uniquenesscontinuestoholdfor(3.5.1)whenallvortexpointsarelocatedsufficientlyclosetotheorigin(“perturbation”fromthesinglevortexsituation)ortoinfinity(“per-turbation”fromthezerovortexsituation)(see[Cho]).Concerningplanarnon-topologicalvortices,itisimportanttounderstandifthelimit,k→0,producesa“concentration”effectonthevortexconfiguration,asitalwaysoccursfortopologicalvortices.Inotherwords,wearenowinterestedinexploringthepossibilityofconstructingasolutionuλfortheproblem⎧⎪⎪N⎨−u=λeu(1−eu)−4πδzj⎪⎪j=1⎩u(x)→−∞as|x|→+∞suchthatNλeuλ(1−euλ)→βjδz,asλ→+∞,(3.5.2)jj=1weaklyinthesenseofmeasure(possiblyalongasequenceλ=λn→+∞),forsuitableβj>0,j=1,...,N.Again,apartialanswercanbefurnishedwhenallthevortexpointscoincide,i.e.,z1=···=zNandβ=8π(N+1)(see(3.4.56)and(3.4.57)),orwhentheyarealldistinctandplacedtoformaZN−symmetricpolygoninR2.Inthiscase,(3.5.2)issatisfiedforanychoiceofβ=β1=···=βN>16π.Finally,allouranalysishasnotyetclarifiedwhetheraproprietyanalogoustothatstatedinTheorem3.2.2fortheMaxwell–HiggsvorticesremainsvalidforChern–Simonsvortices.Namely,Openproblem:Doesanyfiniteenergy(static)solutionoftheChern–Simonsfieldequations(1.2.33)and(1.2.34)inR2reducetoasolutionoftheselfdualequation(1.2.45)? 3.5Finalremarksandopenproblems129Anaffirmativeanswertothisquestionwouldshowthatnomixedvortex-antivortexconfigurationsoccurfortheselfdualChern–Simonsmodeldiscussedabove,inaman-nersimilartowhathappensfortheMaxwell–Higgsmodel.Itwouldbeveryinterestingtoprovideananswertosuchaquestion,wereitlimitedtoChern–Simonssolutionssubjecttotopologicalboundaryconditionatinfinity,orperiodicones(asdiscussedinthefollowingchapter).ConcerningotherChern–Simonsmodels,wementionthattheexistenceofselfdualMaxwell–Chern–Simons–Higgsvortexconfigurations(A,φ,N)satisfying(1.2.63)undertopological-typeboundaryconditionshasbeenestablishedbyChae–Kimin[ChK1].Sincethetopologicalboundaryconditionsrequireanon-vanishingpropertyforthemagnitudeoftheHiggsfieldatinfinity,weseethat(bytakingintoaccount(1.2.59),(1.2.61),and(1.2.62))fortheMaxwell–Chern–Simons–Higgsmodel,thisamountstosatisfyν2|φ|2→ν2andN→,as|z|→+∞;(3.5.3)kwhilenon-topologicalboundaryconditionsareexpressedas|φ|2→0andN→0,as|z|→+∞.(3.5.4)In[ChK1],theauthorssuccededinshowingthatunder(3.5.3)thecorrespondingel-lipticproblem(2.1.14)canbeformulatedvariationallyoverthespaceH2(R2).Thus,inanalogytothe6th-orderChern–Simonsmodel,theyobtainatopologicalsolutionbyaminimizationprocedure.Forthissolution,therateofconvergencein(3.5.3)isexponentiallyfast.Inaddition,theauthorsalsouseaniterationschemetoobtainanalternativeexistenceresultyieldingtoasolutionhavingtheadvantagetopasstothelimitinstrongnormintheabelian–Higgslimit(i.e.,σ→0andqfixed),andinweaknormintheChern–Simonslimit(i.e.,σ→0andq→+∞andk=σfixed).q2Itisinterestingtonotethatananalogousvariationalapproachworksequalywelltotreatplanartopologicalvorticesforthenon-abelianChern–Simonsmodel(1.3.99),(1.3.100),and(1.3.101).Inthiscase,undertheansatz(1.3.116),(1.3.117),thefiniteenergyconditionimposesthatthecomponentφaoftheHiggsfieldsatisfies!|φa|2(ν2−|φb|2Kba)<+∞,(3.5.5)R2∀a=1,...,r,whereristherankofthe(semisimple)gaugegroupwithassociatedCartanmatrixK=(Kba).Henceinthissituation,topologicalvorticesarerequiretosatisfyra|2→|φa|2:=ν2(K−1)|φ0ab,as|z|→+∞,(3.5.6)b=1fora=1,...,r.In[Y6],Y.Yangintroducedavariationalprinciple(intheSobolevspaceH1(R2,Rr)ofvector-valuedfunctions)correspondingtotheellipticsystem(2.1.21), 1303PlanarSelfdualChern–SimonsVorticessubjecttotheboundarycondition(3.5.6).Again,byaminimizationprinciple,theauthorin[Y6]derivesatopologicalnon-abelianChern–SimonvortexinR2,charac-terizedbypropertiesanalogoustothoseestablishedaboveforthetopologicalvortexsolutionofthe6th-orderChern–Simonsmodel(describingthecaseforrankr=1).Wereferto[Y6]or[Y1]fordetails.Observethattheelementφ0=φaEa,with|φa|2in(3.5.6),definesazerofor00the(non-abelian)selfdualpotentialVin(1.3.101).Hence,itdefinesavacuumstateforthesystem,knownastheprincipalembeddingvacuum.However,Vadmitsothervacuastatesφ0=φaEa,wherethecomponentφamayvanish(i.e.,φa=0)for0000somea∈{1,...,r}.Itisaninterestingproblem,completelyopentoinvestigation,todeterminevortexconfigurationsasymptoticallygauge-equivalenttothosevacua,as|z|→+∞.Inthisrespect,eventheextremecasewhereφa=0,∀a=1,...,r,0(knownastheunbrokenvacuumstate)describingthe“non-topological”situation,hasnotbeenhandledyet.Onthecontrary,fortheMaxwell–Chern–Simons–Higgsmodel,non-topologicalvortexsolutionssatisfyingtheboundarycondition(3.5.4)areavailable.TheyhavebeenconstructedbyChae–Imanuvilovin[ChI3]byextendingtotheellipticsystem(2.1.14)theperturbationapproachin[ChI1]discussedabove.However,nothingisknownonwhethertheyadmita“concentration”behavioraroundthevortexpoints(astheChern–Simonsparametertendstozero),animpor-tantinformationformeaningfulphysicalapplications.Finally,abouttheplanartopologicalvortexconfigurationsdiscussedabove(asforthe6th-orderChern–Simonsmodel),itisreasonabletoexpecttheirunique(andpos-siblysmooth)dependenceonthevortexpoints.ButforboththeMaxwell–Chern–Simons–Higgsmodelandthenon-abelianChern–Simonsmodel,suchauniquenessissueisstillunresolved. 4PeriodicSelfdualChern–SimonsVortices4.1PreliminariesWedevotethisChaptertothestudyofperiodicChern–Simonsvortices.Again,wecon-centratemainlyonthe6th-orderChern–Simonsmodel(1.2.45),wherewecancarryoutarathercompleteanalysis.Weshallgiveindicationsofapossiblegeneralizationtoothermodels.Let⊂R2betheperiodiccelldomainin(2.1.27),and{z1,...,zN}⊂betheassignedvortexpoints(repeatedaccordingtotheirmultiplicity).Byconsideringthenormalizationν2=1(alwayspossiblevia(3.1.2)),weareleadtoinvestigatethefollowingellipticproblem⎧⎪⎪N⎨−u=λeu(1−eu)−4πδzin,j(4.1.1)⎪⎪j=1⎩udoublyperiodicon∂,withλ>0givenin(3.1.3).Again,itisconvenienttoworkwithv,theregularpartofu,definedviathedecom-positionu=u0+v,(4.1.2)withu0theuniquesolution(see[Au])oftheproblem⎧⎪⎪N4πN⎨u0=4πδz−in,j||(4.1.3)⎪⎪'j=1⎩u0=0,u0doublyperiodicon∂,with||=Lebesguemeasureon.Thus,intermsofv,theproblemisreducedtosolving:− v=λeu0+v(1−eu0+v)−4πNin,||(4.1.4)vdoublyperiodicon∂. 1324PeriodicSelfdualChern–SimonsVorticesNoticethattheweightfunctioneu0issmooth,doublyperiodicon∂,andvanishesexactlyatz1,...,zNaccordingtotheirmultiplicity.Moreprecisely,
u0(z)=O|z−z2njej|,asz→zj,(4.1.5)fornjthemultiplicityofzjandj=1,...,N.Westartbypointingoutsomeelemen-tarypropertiesvalidforasolutionvof(4.1.4).Firstofall,asanimmediateconsequenceofthemaximumprinciple,wehaveu0+v<0in;(4.1.6)while,afterintegrationover,wefind!λeu0+v1−eu0+v=4πN.(4.1.7)From(4.1.7)weseethat!2λeu0+v−1=λ||−4πN.24Andsowededucethefollowingnecessaryconditionforthesolvabilityof(4.1.4):16πNλ≥.(4.1.8)||Inaddition,ifwewrite!!v=w+d,withw=0andd=−v;(4.1.9)thenby(4.1.6)and(4.1.7),wefind4πNd||'≤e≤',(4.1.10)λeu0+weu0+w4πN|| w||L∞()=|| v||L∞()≤λ+.||Therefore,well-knownellipticestimatesimplythatwisboundedin,anduniformlyso,foreveryλinaboundedsubsetoftheinterval[16πN,+∞).SinceJensen’sinequal-'||ity(see(2.5.8))impliesthat−eu0+w≥1,weseethatthesameuniformboundednesspropertyholdsforv.Infact,afterabootstrapargument,suchapropertycanbeex-tendedtoholdinCm()-norm,foranygivenm∈Z+.Inotherwordsthefollowingholds:Lemma4.1.1(a)Ifproblem(4.1.4)admitsasolution,thenλ≥16πN.||(b)Foragivenλ16πN0>,thesetofsolutionsof(4.1.4)withλ∈[0,λ0]iscompact||inC2,α()α∈(0,1),andinanyotherrelevantspace. 4.1Preliminaries133Inthefollowingweshallbeinterestedindescribingtheasymptoticbehaviorofthesolutionfor(4.1.4)asλ→+∞.Forthisreason,wepointoutalsothefollowingestimate(uniforminλ)intheweakernorm.Lemma4.1.2Letλ>0andvbeasolutionfor(4.1.4).Foranyq∈(1,2)thereexistsaconstantCq>0(independentofλ)suchthat||∇v||Lq()≤Cq.Remark4.1.3AsaconsequenceofLemma4.1.2andSobolev’sembeddingtheorem,weknowthat,foranysequenceλn→+∞,ifvλsolves(4.1.4)withλ=λn,then'n(alongasubsequence)wn=vλ−−vλconvergesinLp(),∀p≥1andpointwisennalmosteverywhere.'qProofofLemma4.1.2.Setw=v−−v,andforq∈(1,2)letp=beitsdualq−1exponent.WeconsidertheSobolevspace1,p2Hp()=v∈W(R):vdoublyperiodic,withperiodiccelldomainloc(4.1.11)=W1,pR2
a1Z×a2Z;andrecallthat||∇v||Lq()=||∇w||Lq()(!!)=sup∇w∇φ:φ∈Hp()||∇φ||Lp()=1andφ=0.Sinceforp>2,wehavethecontinuousSobolevembedding:Hp()→L∞(),'thenforφ∈Hp,φ=0and||∇φ||Lp()=1,wefind||φ||L∞()≤CforasuitableconstantC>0.So,bymeansof(4.1.6)and(4.1.7),wecanestimate!!!!∇w·∇φ=− w·φ=− v·φ=λeu0+v(1−eu0+v)φ!u0+vu0+v≤||φ||L∞()λe(1−e)≤4πNC,andthedesiredestimatefollows.Concerningthestructureofthesolution-setof(4.1.4)(forλ>0large),weseethatbythecondition(4.1.7),wearestillleadtoexpecttwoclassesofsolutions.Namely,thosesatisfyingeu0+v→1a.e.in,asλ→+∞,(4.1.12) 1344PeriodicSelfdualChern–SimonsVorticeswhich,inanalogytotheplanarcase,weshallcallofthe“topological-type”;andthosesatisfyingeu0+v→0a.e.in,asλ→+∞,(4.1.13)whichwecallofthe“non-topological-type.”Wedevotetherestofthischaptertoconstructingsolutionsto(4.1.4)foreachsuchtype.4.2Constructionofperiodic“topological-type”solutionsWetakep=2in(4.1.11),thenHp=2()reducestotheSobolevspaceH()definedin(2.4.23),andfurnishesthenaturalspaceinwhichtoseeksolutionsfor(4.1.4).Afirstcontributiontowardstheexistenceofsolutionsfor(4.1.4)(forλ>0large)wasprovidedbyCaffarelli–Yangin[CY]bymeansofasub/supersolutionmethod.Tothispurpose,recallthatafunctionv∈H()iscalleda(weak)subsolution(respectivelysupersolution)for(4.1.4)if∀ϕ∈H(),withϕ≥0inwehave:!!∇v·∇ϕ−λeu0+v(1−eu0+v)ϕ≤0(resp.≥0).(4.2.1)Lemma4.2.4Thereexistsλ∗>0andasmoothfunctionv−∈H()suchthatv−definesasubsolutionfor(4.1.4),∀λ>λ∗.Proof.Weshallconstructv−asasmoothdoublyperiodicfunctionon∂satisfying− v≤λeu0+v(1−eu0+v)−4πNin,(4.2.2)||forlargevaluesofλ.Tothispurpose,letε>0besufficientlysmallsothatB2ε(zj)⊂,j=1,...,Nandforzj=zk,B2ε(zj)∩B2ε(zk)=∅.Considerasmoothcut-offfunctionfε∈C∞()suchthat0≤fε≤1,fε=1in∪NBε(zj)whilefε(z)=0,∀z∈/0j=1∪NB2ε(zj).Setj=1!8πNgε=fε−−fε.||Letv−0betheuniquesolution(dependingonε)fortheproblem: v−0=gεin,'v−0∈H()andv−0=0.Setv0−d,(4.2.3)−=v− 4.2Constructionofperiodic“topological-type”solutions135withd>0sufficientlylargetoguaranteeu0+v−<0in.(4.2.4)'Notethat−fε=O(ε2),andtherefore∀z∈∪NBε(zj),wehave:j=1

8πN24πNgε(z)=1+Oε≥,forε>0sufficientlysmall.(4.2.5)||||Ontheotherhand,by(4.2.2)itfollowsthat
µε:=max{eu0+v−(z)u0+v−(z)−1,∀z∈∪NB0ej=1ε(zj)}<0,andwefind
λeu0(z)+v−(z)eu0(z)+v−(z)−1+4πN≤−λ|µε|+4πN,(4.2.6)||0||∀z∈∪NBε(zj).Thus,ifwecombine(4.2.5)and(4.2.6),weseethatifwefixj=1ε>0sufficientlysmall,thenwefindλ∗>0suchthatu0+v−u0+v−4πNλee−1+≤gεin,∀λ≥λ∗,||and(4.2.2)followswithv−in(4.2.3).Remark4.2.5Byacloserlookattheargumentsabove,weseethattheparameterλ∗dependsonthevalue:min|zj−zk|,(besideobviouslyNand)butnotontheactualzj=zkpositionofthevortexpoints.Toobtainasolutionfor(4.1.4),wecanproceedeitherbyaniterationscheme(see[CY]and[Y1]),orvariationallybymeansofaminimizationproblemoverH().Weshallpresentthevariationalapproachtoremaininthesamespiritoftheplanartopologicalvortexproblem(discussedinSection3.1).Also,thevariationalapproachwillbeusefulinidentifyingthesecondclassof“non-topological-type”solutions.Tothispurpose,let!!!12λu0+v24πNJλ(v)=|∇v|+e−1+v,v∈H().(4.2.7)22||Byvirtueof(2.4.24),weseethatJλiswell-defined,continuoslyFrechetdifferentiable,´andweaklylowersemicontinuousinH().Furthermore,itscriticalpointscorrespondto(weak)solutionsto(4.1.4).Lemma4.2.6Letv−∈H()beasubsolutionfor(4.1.4)andset={v∈H():v≥v−a.e.in}.ThenJλisboundedfrombelowinwhereitsinfimumisattainedatacriticalpoint. 1364PeriodicSelfdualChern–SimonsVorticesProof.Clearly,!infJλ≥4πN−v−.(4.2.8)Toobtainthattheinfimumin(4.2.8)isachieved,weconsideraminimizingsequencevn∈;thatis!!!12λu0+vn24πNJλ(vn)=|∇vn|+e−1+vn→infJλ.22||Therefore,!!||∇vn||L2≤Candv−≤vn≤C,forasuitableconstantC>0.Inotherwords,vnisuniformlyboundedinH(),andsowecanfindasubsequence(denotedinsomeway):vn→vweakinH()andpointwisea.e.in.Consequently,v∈andbytheweaklylowersemicontinuityofJλ,wefindJλ(v)=infJλ.λSovgivesthedesiredminimumofJλin.ThefactthatvdefinesacriticalpointforJλinH()isaconsequenceofawell-knownpropertyofasubsolution(supersolution)inavariationalguise(seee.g.,TheoremI.24in[St1]ortheappendixof[T1]orLemma5.6.3in[Y1]).So,afterLemma4.2.4andLemma4.2.6,weconcludetheexistenceofasolutionforproblem(4.1.4),providedthatλ>0issufficientlylarge.Inotherwords,setting={λ>0:problem(4.1.4)admitsasolution},(4.2.9)wethenknowthatcontainsaninfiniteinterval,and(recalling(4.1.8)),thatitisawell-definedvalue:16πNλc:=inf{λ,λ∈}≥.(4.2.10)||Lemma4.2.7Wehave=[λc,+∞),(4.2.11)andso,problem(4.1.4)admitsasolutionifandonlyifλ≥λc.Proof.Letλ1∈,sothatproblem(4.1.4)withλ=λ1admitsasolutionv1.Byvirtueof(4.1.6),v1definesasubsolutionfor(4.1.4)foreveryλ>λ1.Thus,wecanapplyLemma4.2.6(withv−repacedbyv1)toobtainasolutionof(4.1.4)foreveryλ>λ1. 4.2Constructionofperiodic“topological-type”solutions137Inotherwords,λ1∈impliesthatthewholeinterval[λ1,+∞)⊂.Thisallowustoconcludethat(λc,+∞)⊂.Finally,wecanalsoderiveasolutionforλ=λcbyalimitingargumentonthebasisofLemma4.1.1(b).Itisusefultoobservethatforanyfixedλ∈[λc,+∞),byLemma4.2.6,wecanconstructamaximalsolutionv1,λfor(4.1.4)justbysetting:v1,λ(z)=sup{v(z):v∈H()isasubsolutionfor(4.1.4)}=max{v(z):visasolutionfor(4.1.4)}<−u0(z).Inparticular,v1,λismonotoneincreasinginλ;thatis,ifµ<λ,thenv1,µ(z)λcλ→λ+cInconclusion,wehaveproventhefollowingresult:Theorem4.2.8Thereexistλ16πNsuchthatc≥||(i)foreveryλ≥λc,problem(4.1.4)admitsamaximalsolutionv1,λwhichismonotonicallyincreasinginλ.Thatis,ifλc≤λ<µ,thenv1,λ(x)λc(ii)Forλ<λc,problem(4.1.4)admitsnosolutions.Remark4.2.9Concerningthedependenceofλconthepositionofvortexpointsz1,...,zN,wecantakeintoaccountRemark4.2.5toseethatbesidesandN,itsvaluedependsonlyon(thereciprocalof)theminimaldistancebetweendistinctvortexpointsandnotontheiractuallocations.Soa“discontinuity”forλc(z1,...,zN)canoccuronlywhendifferentvorticescollapsetogether.SettingSλ={v∈H():visasolutionfor(4.1.4)},(4.2.12)wehavethenshownthatthesetSλ=∅ifandonlyifλ≥λc,inwhichcaseitalwaysadmitsamaximalelement.Remark4.2.10Forλ>λc,itisalsopossibletoconstructaminimalsolutionωλamongallpossiblesolutionsof(4.1.4)biggerthanthesubsolutionv1,λ.Tothispur-cpose,itsufficestoconsiderthesupersolution,+(z)=minv(z):v∈Svλandv≥v1,λin,λc 1384PeriodicSelfdualChern–SimonsVorticesandasinLemma4.2.6,obtainωλastheminimizerofJλontheset{v∈H():v≤v≤v+a.e.in}.Asamatteroffact,setting1,λcλλ={v∈H():v1,λ≤v≤ωλa.e.in},(4.2.13)cweseethattheminimalsolutionωλischaracterizedbythefollowingproperties:Jλ(ωλ)=infJλ,(4.2.14)λandSλ∩λ={ωλ},(4.2.15)∀λ>λc.Byarecentresultoftheauthor(see[T7]),weactuallyknowthatthemaximalandminimalsolutionisoneandthesameforsufficientlylargeλ>0.Next,weexploitfurtherthevariationalformulationaboveinordertoshowthatforeveryλ>λc,problem(4.1.4)admitsasolutionotherthanthemaximalone.Tothispurpose,wefirstneedtoderiveanimprovementtothestatementofLemma4.2.6basedontheobservationofBrezis–Nirenberg[BN],assertingthat,forgeneralvariationalprinciplesofellipticnature,minimizationinC1-normisequivalenttomin-imizationinH1-norm.Letv∗=v1,λbethemaximalsolutionfor(4.1.4)atλ=λc.Sinceitdefinesacstrictsubsolutionfor(4.1.4)foreveryλ>λc,wecanuseitinLemma4.2.6toobtainasolutionvλto(4.1.4)satisfying:Jλ(vλ)=inf{Jλ(v):v∈H()andv≥v∗,a.e.in}.(4.2.16)Notethat,bythestrongmaximumprinciple,wehavethestrictpointwiseinequality:vλ(x)−v∗(x)>0,∀x∈.(4.2.17)Therefore,vλdefinesalocalminimumforJλinC0()-topology.Thefollowingstrongerresultholds:Lemma4.2.11vλdefinesalocalminimumforJλinH().Proof.Wearguebycontradictionandassumethatforeveryρ>0thereexistsv∈H():||v−vλ||≤ρandJλ(v)v∗in,fornsufficientlylarge.Butthisisclearlyimpossiblesince(4.2.18)wouldcontradict(4.2.16).IfwecombineLemma4.2.11withthefactthatthefunctionalJλisunboundedfrombelowinH(),weseethatJλadmitsa“mountain-pass”geometryinthesenseof(2.3.3)(cf.[AR])andwecandrawthefollowingconclusion:Proposition4.2.12Foreveryλ>λc,thefunctionalJλadmitsatleasttwocriticalpointsinH().AsafirststeptowardstheproofofProposition4.2.12,weneedtoestablishthefollowing.Lemma4.2.13ThefunctionalJλsatisfiesthe(PS)-conditionatanylevelc∈R.Proof.Actually,wecheckthefollowingstrongerproperty:if{vn}⊂H()satisfiesJ(vn)→0inH∗()("H()),thenvnadmitsastronglyconvergentsubsequence.λTothispurpose,write!!vn=wn+tn:tn=−vnandwn=0.Wehave!(vu0+vneu0+vno(1)=Jλn)(1)=λe−1+4πN,!(4.2.22)(v2u0+vnu0+vno(||wn||)=Jλn)(wn)=||∇wn||2+λee−1wn. 1404PeriodicSelfdualChern–SimonsVorticesUsingHolder’sinequality,fromthefirstpropertyin(4.2.22),wederive!!12(u+v124πNe0n)−||2e2(u0+vn)+≤o(1),λyieldingtotheuniformestimate!e2(u0+vn)≤C,(4.2.23)forasuitableconstantC>0.Ontheotherhand,bymeansofJensen’sinequality(2.5.8),!!eu0+wn≥exp−u−0+wn=1,wecanuse(4.2.23)toobtaintheupperbound:!!!12etn≤etn−eu0+wn=−eu0+vn≤−e2(u0+vn)≤C.(4.2.24)Atthispoint,bythesecondpropertyin(4.2.22),wededucetheestimate:!o(||w2u0+vnu0+vnn||)=||∇wn||2+λe(e−1)wnL()!=||∇wn||2+λeu0+vn(eu0+wn+tn−eu0+tn)wn(4.2.25)L2()!+λeu0+tneu0+vnw2n≥||∇wn||L2()−C||wn||L2(),∀n∈N.Consequently,||∇wn||L2()≤C,(4.2.26)forasuitableconstantC>0.Inparticular,bymeansofMoser–Trudingerinequality(2.4.24),foreveryp≥1,wehave!eu0+wn≤Cp,(4.2.27)Lp()∀n∈N,withasuitableconstantCp>0(dependingonlyonp).Observingthatthefirstidentityin(4.2.22)alsoimplies!etneu0+wn≥4πN+o(1),λwecanuse(4.2.27)(withp=1)todeducethelowerbound4πNtn≥log−logC1+o(1).λ 4.2Constructionofperiodic“topological-type”solutions141Combiningtheestimatesabovewith(4.2.24),weconcludethat|tn|≤C,(4.2.28)∀n∈NandforasuitableC>0.Since(4.2.26)and(4.2.28)implythatvnisuni-formlyboundedinH(),wecanfindasubsequence(denotedinthesameway)toassertthatv2(),nvweaklyinH()andstronglyinLforasuitablev∈H().Furthermore,!2=J(vu0+vnu0+vn||∇(vn−v)||L2λn)(vn−v)−λe(e−1)(vn−v)!!−4πN−(vn−v)+∇v·∇(vn−v)≤o(1)||v||e2(u0+vn)||(u0+vn)||4πNn−v||+λL2+||eL2+1||vn−v||L2()||2+o(1)≤C||vn−v||L2()+o(1),asfollowsfrom(4.2.26),(4.2.27),and(4.2.28).Consequently,||vn−v||→0,andthedesiredconclusionfollows.Wearenowreadytogive:ProofofProposition4.2.12.Letρ0>0besuchthatJλ(vλ)=infJλ(v),||v−vλ||≤2ρ0withvλasgiveninLemma4.2.11.IfJλ(vλ)=inf||v−vJλ(v),thenwewouldfindλ||=ρ0alocalminimumforJλin∂Bρ(vλ)toprovideuswithasecondcriticalpoint.Hence0supposethatJλ(vλ)0sufficientlylargetosatisfyt0>ρ0andJλ(vλ−t0)Jλ(vλ)γ∈Pt∈[0,1]definesacriticalvalueforJλ(see[AR]and[St1]TheoremII.6.1).Sincecλ>Jλ(vλ),cλyieldstoacriticalpointdifferentfromvλ.AsaconsequenceofProposition4.2.12wehave:Theorem4.2.14Forλ>λc,problem(4.1.4)admitsasecondsolution(distinctfromthemaximalsolution).Ournextgoalistoobtainmorepreciseinformationabouttheasymptoticbehaviorofthosedifferentsolutions,asλ→+∞.Moreprecisely,wewishtoknowwhether,respectively,theysatisfythe“topological-type”conditionof(4.1.12)andthe“non-topological-type”conditionof(4.1.13).Acriterionthatdistinguishesbetweenthesetwoclassesofsolutionsisgivenasfollows:Lemma4.2.15Letvλbeasolutionto(4.1.4),andset!dλ=−vλ.(1)Iflimsupλ→+∞λedλ<+∞,asλ→+∞,theneu0+vλ→0,inLp()∀p≥1andpointwisea.e.in.(4.2.31)(2)Ifλedλ→+∞,asλ→+∞(orpossiblyalongasequenceλn→+∞),thenasλ→+∞(oralongthegivensequence):p()∩Cm({z(i)u0+vλ→0inL1,...,zN}),(4.2.32)locforeveryp≥1,andm∈Z+;N(ii)λeu0+vλ(1−eu0+vλ)→4πδz=4πnjδz,(4.2.33)jjj=1j∈Jλ(1−euλ)2→4πn2δz,(4.2.34)jjj∈Jλ(1−euλ)→4πnj(1+nj)δz,(4.2.35)jj∈Jweaklyinthesenseofmeasurein.WhereJ⊂{1,...,N}isasetofindicesiden-tifyingalldistinctvorticesin{z1,...,zN},andnj∈Nisthemultiplicityofzjforj∈J. 4.2Constructionofperiodic“topological-type”solutions143Proof.Toestablish(4.2.31),noticethatbythegivenassumption,necessarilydλ→−∞asλ→+∞.Asaconsequence,weclaimthateu0+vλ→0pointwisea.e.in.(4.2.36)Indeed,onthebasisofRemark4.1.3weknowthat,foranysequenceλn→+∞,!wpn=vλ−dλ=vλ−−vλ→w0,inL()andpointwisea.e.innnnn(bytakingasubsequenceifnecessary).Consequently,eu0+vλn=edneu0+wn→0pointwisea.e.in,foranysequenceλn→+∞.Thuswededuce(4.2.36).Atthispointby(4.1.6),wecanusedominatedconvergencetoconcludethateu0+vλ→0inLp(),∀p≥1asclaimed.Toestablish(4.2.32)and(4.2.36),westartbyobservingthatthegivenassumptionimpliesthat,dp(),p≥1;(4.2.37)λ→0andu0+vλ→0inLasλ→+∞(oralongthegivensequenceλn→+∞).Tothispurpose,noticethatby(4.1.6)wehaveedλ∈(0,1),∀λ>0.Assumethatliminfedλ:=A∈[0,1].(4.2.38)λ→+∞Thenalongasequenceλk→+∞,wefind!dλpek→Aandwk=vλ−−vλ→w0inL()∀p≥1,andpointwisea.e.in.kkNotice,inparticular,that!w0=0.(4.2.39)Ontheotherhand,ifweuse(4.1.7),then!eu0+wk(1−eu0+vλk)=4πN→0,ask→+∞.dλλkekSowecanuseFatou’slemmatoconcludethatAeu0+w0=1.HenceA>0,andsince1u0+w0=log,Awecanuse(4.2.39)tofind!1log=−u0+w0=0,thatisA=1.AConsequentlydλ→0,asλ→+∞(oralongthegivensequence),andu0+vλ→0inLp(),∀p≥1. 1444PeriodicSelfdualChern–SimonsVorticesWiththisinformation,wecanproceedexactlyasintheproofofProposition3.2.9,toobtaintheestimates!Nλeu0+vλ(1−eu0+vλ)ϕ−4πϕ(zj)≤|| ϕ||∞||u0+vλ||1(4.2.40)L()L()j=1∀ϕ∈H()∩C∞,andforδ=∪NBδ(zj),δ>0,j=1u0+vλu0+vλCδ(p)||u0+vλ||L∞(δ)+||λe(1−e)||L∞(δ)≤(4.2.41)λforanygivenp≥1,withCδ(p)>0asuitableconstantindependentofλ.Thus,from(4.2.40)wereadilyget(4.2.33);whilewecancompletetheproofof(4.2.32),byvirtueof(4.2.41)andabootstrapargument.Finally,byarguingasintheproofofCorollary3.2.10,wecanuseaPohozaev-typeidentityaroundeachvortexpointzj,j=1,...,Ntodeduce(4.2.34)and(4.2.35).ByLemma4.2.15wecanimmediatelyclassifythemaximalsolutionv1,λ,asbeingofthe“topological-type.”Indeed,!!forλ≥λc,d1,λ=−v1,λ≤−v1,λ:=d1,λ;ccandtherefore,λed1,λ→+∞.Thusweconclude:Corollary4.2.16Themaximalsolutionof(4.1.4)(asgivenbyTheorem4.2.8)satis-fiesthe“topological-type”condition(4.1.12),wheretheconvergenceholdsinLp()∀p≥1andpointwisea.e.in.Inaddition,itsatisfiestheconvergenceproperties(4.2.32),(4.2.33),(4.2.34),and(4.2.35).4.3Constructionofperiodic“non-topological-type”solutionsAtthispoint,itisnaturaltoaskwhetherthe“mountain-pass”solutionconstructedaboveisofthe“non-topological-type.”Afirstanswertothisquestionhasbeenpro-videdbyDing–Jost–Li–Pi–Wangin[DJLPW]withanapproachthatworksequallywellinahigherdimensionandallowsustotreatvortex-typesolutionsfortheSieberg–Wittenfunctional(cf.[Jo]).InourcontexttheirresultcompletesTheorem4.2.14asfollows:Theorem4.3.17[DJLPW]Forλ>0sufficientlylarge,problem(4.1.4)admitsa“non-topological-type”solutionvλ,inthesensethat(4.2.31)holds.BeforegoingintothedetailsoftheproofofTheorem4.3.17,wementionthatprevi-ousresultsrelativeto“non-topologicaltype”solutionsarecontainedin[T1],[DJLW2],[DJLW3],and[NT3].Thoseresultsconcernmainlythesingleordoublevortexcase(i.e.,N=1,2),andhavetheadvantagetoyieldsolutionsthatverifytheconvergencepropertyin(4.2.31)withrespecttoC0()−norm.Weshallreturntodiscussthisas-pectinSection4.4. 4.3Constructionofperiodic“non-topological-type”solutions145Asalreadymentioned,theproofofTheorem4.3.17reliesina“mountain-pass”construction.Howevertoattainthedesirednon-topologicalinformation,weneedtoreplacethelocalminimumvλofLemma4.2.11withtheminimalsolutionωλ,aschar-acterizedby(4.2.14)and(4.2.15).ButwecancheckthatωλdefinesalocalminimumonlywithrespecttothestrongerW2,2R2
a1Z×a2Z-topology.Therefore,wealsoneedtomodifythepseudogradientdeformationflowin(2.3.6),(2.3.7)(cf.[St1],[R])withthemoreregularheatflow:⎧⎪⎨vt= v+λeu0+v(1−eu0+v)−4πNin×R+,||v(·,0)=g0,(4.3.1)⎪⎩v(·,t)doublyperiodicon∂,∀t≥0,wheretheinitialdatag0issuitablychosen.Tobemoreprecise,let
v∈W2,22
aXλ=R1Z×a2Z:v≤ωλin⊂H().ThenXλisaclosedconvexsubsetofW2,2R2
a1Z×a2Z,withinducednormdenotedby||·||X.λSincewλisasolutionfor(4.1.4),weknowthatXλisinvariantundertheheatflow(4.3.1),andthefollowingholds:Lemma4.3.18Foreveryg0∈Xλ,problem(4.3.1)admitsauniquesmoothsolutionvin×(0,+∞),suchthatv(·,t)∈Xλ,whichdependscontinuouslyontheinitialdatag0,∀t∈(0,+∞).Furthermore:(i)Themap:t→(||v(·,t)||Xλ,||vt(·,t)||L2())iscontinuousin[0,+∞)and||v(·,t)−g0||X→0,ast→0+.λ(ii)Ifsup||v(·,t)||L2()+||vt(·,t)||L2()<+∞,(4.3.2)t>0thenast→+∞,v(·,t)convergesinXλtoasolutionfor(4.1.4).Proof.Bygenerallocalexistenceresultsfornonlinearparabolicequations(cf.[LSU]and[Fre]),weknowthatforg0∈XλthereexistsT>0suchthat(4.3.1)admitsasolutionv∈L2(0,T;Xλ)∩L∞(0,T;H())andvt∈L2(0,T;L2()).Sincev(·,t)∈Xλ,thenu0(z)+v(z,t)<0,∀z∈and∀t∈[0,T].Therefore,setting
f(z,t)=λeu0(z)+v(z,t)1−eu0(z)+v(z,t)−4πN,(4.3.3)||wefind4πN∂f||f||L∞(×[0,T])≤λ+and≤3λ|vt|in×[0,T].(4.3.4)||∂t 1464PeriodicSelfdualChern–SimonsVorticesByfamiliararguments,wederivetheestimates:1d214πN||vt(·,t)||2≤||2λ+||v(·,t)||L2()(4.3.5)2dtL()||1d222dt||v(·,t)||L2()≤3λ||vt(·,t)||L2(),∀t∈[0,T].(4.3.6)Thus,bymeansofaGronwalltypeinequality,from(4.3.5)and(4.3.6),wededuce:
2+||v(·,t)||2≤c3λt||vt(·,t)||L2()L2()λ1+e||g0||Xλ,∀t∈[0,T],(4.3.7)withcλ>0aconstantindependentoft.Inturn,byellipticestimates,wefind
2+||v(·,t)||3λt||∇vt(·,t)||L2()Xλ≤Cλ1+e||g0||Xλ,∀t∈[0,T],(4.3.8)withCλasuitableconstantdependingonλonly.Inparticular,vt∈L2(0,T;H())andwededuce(i)bymeansofwell-knownre-sults(seee.g.,[Ev]and[LSU]).Furthermore,wecanuseabootstrapargumenttogetherwithstandardparabolicestimatestoderiveestimatesinHolderspacesanalogousto(4.3.8).Inthisway,wecancheckforsmoothnessofthesolutionanduseacontinu-ationargumenttoconcludethatnecessarilyT=+∞.Sovisgloballydefinedandsmoothin×(0,+∞)asclaimed.Moreoverif(4.3.2)holds,thenwecanreplacetheright-handsideof(4.3.7)withaconstantindependentoft,andinturnobtaincor-respondingestimatesforvandvtinHolderspacestoholduniformly,∀t>0.Thisfactallowsustoshowconvergenceofv(·,t)inXλ,ast→+∞,towardsalimitingfunctionthatmustsatisfy(4.1.4).Similarly,wecanestimatethedifferencebetweentwosolutionsv1andv2satisfy-ing(4.3.1)intermsoftheirinitialdatag1andg2∈Xλ,respectively.Thus,wefindthat
≤C3λt||v1(·,t)−v2(·,t)||Xλe+1||g1−g2||X,∀t>0,(4.3.9)λλforsuitableCλ>0dependingonλonly.Thisprovesuniquenessandcontinuousdependenceofthesolutionontheinitialdata.OnthebasisoftheinformationinLemma4.3.18weobtain:Lemma4.3.19TheminimalsolutionωλdefinesastrictlocalminimumforJλinXλ.Proof.Firstofallobservethat,ifv=v(z,t)isasolutionof(4.1.4),thenthefunctionalJλ(v(·,t))ismonotonedecreasingint,andwehave!d22Jλ(v(·,t))=−|vt(x,t)|dx=−||vt(·,t)||2.(4.3.10)dtL() 4.3Constructionofperiodic“non-topological-type”solutions147Wearguebycontradictionandsupposethatforeveryδ>0small,wehavewδ∈Xλ:||ωλ−wδ||X=δandJλ(wδ)≤Jλ(ωλ).(4.3.11)λRecallthatωλ−v1,λ>0in.Soforδ>0sufficientlysmall,wecanuseLemmac4.3.18,tofindTδ>0small,suchthat,ifvisthesolutionof(4.3.1)withv(·,0)=wδ,thenδv1,λ0,t∈[0,Tδ]L()andby(4.3.10)and(4.3.11)weobtain!Tδ||v2Jλ(v(·,Tδ))≤Jλ(wδ)−t(·,t)||2≤Jλ(ωλ)−ε0Tδ0andε0>0suchthat∀v∈Xλ:||v−ωλ||X=ρ0thenJλ(v)≥Jλ(ωλ)+ε0.(4.3.12)λSinceJλ(ωλ−s)→−∞ass→+∞,wecanfixs0>0sufficientlylargesothatJλ(ωλ−s0)≤minJλ−1.(4.3.13)||v−ωλ||Xλ≤ρ0Foranys∈[0,s0]andt≥0,weconsiderthetwo-parameterfamilyoffunctionsv(z,t,s)suchthatv(·,·,s)istheuniquesolutionfor(4.3.1)withv(·,0,s)=ωλ−s.Observethatv(·,t,0)=ωλ,∀t≥0,andby(4.3.10),Jλ(v(·,t,s0))≤Jλ(v(·,0,s0))=Jλ(ωλ−s0).Therefore,by(4.3.13),thefunctionv(·,t,s0)mustlieoutsidetheballinXλwithcenterωλandradiusρ0.Hence,byacontinuityargument,∀t>0theremustexistst∈(0,s0)suchthat||v(·,t,st)−ωλ||X=ρ0,λandso,Jλ(v(·,t,st))≥Jλ(ωλ)+ε0. 1484PeriodicSelfdualChern–SimonsVorticesNow,alongasequencetn→+∞wemaysupposethatst→s∗,andbythenmonotonicityproperty(4.3.10),wefindJλ(v(·,t,st))≥Jλ(v(·,tn,st))≥Jλ(ωλ)+ε0,nnforanyt∈(0,+∞)andnsufficientlylarge.Passingtothelimitasn→+∞,weconclude:Jλ(v(·,t,s∗))≥Jλ(ωλ)+ε0,∀t∈[0,+∞).(4.3.14)Thisimpliesthats∗>0and!t||v2t(·,τ,s∗)||2dτ=Jλ(ωλ−s∗)−Jλ(v(·,t,s∗))≤C1,∀t>0;(4.3.15)L()0withasuitableconstantC1>0independentoft.Consequently,wecanuse(4.3.6)todeduce!t2≤||ω2||vt(·,t,s∗)||L2()λ−s∗||Xλ+||vt(·,τ,s∗)||L2()dτ0≤C2,∀t>0,(4.3.16)withasuitableconstantC2>0independentoft∈[0,+∞).Atthispoint,bymeansoftheequation(4.3.1),wefind!!2λeu0+v1−eu0+v−v||∇v(·,t,s∗)||2=t(v−−v)L()!1≤(λ||2+C2)||v−−v||L2(),andbyPoincare’sinequalitywederivetheuniformestimate||∇v(·,t,s∗)||L2()≤C3,∀t>0(4.3.17)withasuitableconstantC3>0,independentoft.Moreover,!!112λu0+vλ2−v(·,t,s∗)=Jλ(v(·,t,s∗))−||∇v(·,t,s∗)||2−1−e4πN2L()2112λ≥(Jλ(ωλ)+ε0−C3−||);4πN22andrecallingthatv(·,t,s∗)<ωλin,weobtain!−v(·,t,s∗)≤C,∀t>0;(4.3.18)withasuitableconstantCindependentoft.So,bycombining(4.3.16),(4.3.17),and(4.3.18),wecheckthevalidityofassumption(4.3.2)forv(·,t,s∗).Therefore,bylet-tingt→+∞,weconcludethatv(·,t,s∗)→v2,λinXλ 4.3Constructionofperiodic“non-topological-type”solutions149(andinanyotherrelevantnorm)withv2,λasolutionfor(4.1.4)whichsatisfiesv2,λ≤ωλin.(4.3.19)SinceJλ(v2,λ)=limJλ(v(·,t,s∗))≥Jλ(ωλ)+ε0,t→+∞weseethatnecessarilyv2,λ=ωλ,andso(4.3.19)holdswithastrictinequality.Fur-thermore,property(4.2.15)impliesthatv2,λ∈/λ(definedin(4.2.13)).(4.3.20)Next,weuse(4.3.20)toshowthat!vd2,λd2,λ:=−2,λsatisfieslimsupλe<+∞,(4.3.21)λ→+∞andsobyLemma4.2.15,wecanassertthatv2,λadmitsthe“non-topological-type”behavior(4.2.31)asclaimed.Toestablish(4.3.21),wearguebycontradictionandassumethatalongasequenceλn→+∞,λned2,λn→+∞,asn→+∞.Therefore,wecanapplypart(2)ofLemma4.2.15tovn=v2,λtoseethat,u0+vn→0inC0({z1,...,zN}).Sincenlocu0+v1,λ<0in,forε>0sufficientlysmallwefindnε∈N:∀n>nεwehave:cvNn>v1,λin∪Bε(zj).(4.3.22)cj=1Settingε=minvmj,nn,j=1,...,N,||z−zj||=εforn≥nε,wecanfurtherassumethatmε≥−maxuj,n0+1.||z−zj||=εSo,ifnj∈Nisthemultiplicityofthevortexzj,j=1,...,N;thenε1mj,n≥2njlog−C,(4.3.23)εforeveryn≥nε,withC>0asuitableconstantindependentofε>0,n∈Nandj∈{1,...,N}.Weobservethat− (vn−mε−πN|z−zj|2)=λeu0+vn(1−eu0+vn)≥0,inBε(zj),
j,nvn−mε−πN|z−zj|2|∂Bε(z)≥−πNε2,j,njandbythemaximumprinciplewegetvε2n≥mj,n−πNεinBε(zj),∀j=1,...,N. 1504PeriodicSelfdualChern–SimonsVorticesThusbyvirtueof(4.3.23),forε>0sufficientlysmallandn∈Nsufficientlylarge,wecanalsocheckthatvn>v1,λinBε(zj)∀j=1,...,N.(4.3.24)cPuttingtogether(4.3.22)and(4.3.24),weseethatvn>v1,λin,andsov2,λ∈λ,cnnincontradictionwith(4.3.20).FinalRemarks:Therehasbeenarecentdevelopmentaboutthe“topological-type”andthe“non-topological-type”solutionsfor(4.1.4).In[T7],theauthorhasprovedthatforλ>0sufficientlylarge,themaximalsolutionv1,λistheonlysolutionof(4.1.4)satisfying:'−vλe→+∞asλ→+∞.Inotherwords,ifvnisasolutionfor(4.1.4)withλ=λn→+∞,and'−vnλne→+∞,asn→+∞,thenvn=v1,λforlargen∈N.ThisresultfollowsfromargumentsnsimilartothosepresentedintheproofofTheorem3.3.14andTheorem3.3.15,andwereferto[T7]fordetails.Asaconsequence,wecanuseLemma4.2.15toconcludethatforλ>0large,themaximalsolutionv1,λistheonly“topological-type”solutionfor(4.1.4)(inthesenseof(4.1.12)).Onthisbasis,wenolongerneedthe(howeverinteresting)constructionaboveinordertoobtaina“non-topological-type”solution,aswenowcanclaimsuchexistencedirectlyfromTheorem4.2.14,asgivenbythe“mountain-pass”solutions.Recently,Choein[Cho1]hasobtainedaverydetaileddescriptionoftheasymptoticbehavior(asλ→+∞),ofsuch“mountain-pass”solution.Seealso[ChoK]forothermultiplicityresults.4.4AnalternativeapproachOurnextgoalistoillustrateadifferentconstructionof“non-topological-type”solu-tionswhichenablesustochecktheconvergencepropertyin(4.2.31)withrespecttotheuniformtopology.Thisalternativeapproachwasproposedin[T1]andsubsequentlypursuedin[NT1]and[NT3].Theprocedurepresentedbelowwasfirstintroducedin[CY]tohandlethesimplersituationof“topological-type”solutions(seealso[Y1]).Tobemoreprecise,forv∈H(),weusethedecomposition:!!v=w+d,withd=−vandw=0.Observethat,ifvisasolutionfor(4.1.4),thendsatisfiestheequation!!e2de2(u0+w)−edeu0+w+4πN=0,λ 4.4Analternativeapproach151whosesolvabilityimposesthecondition!!2e2(u0+w)≤λeu0+w,(4.4.1)16πNfromwhichwecanalsore-establishthenecessaryconditionof(4.1.8).Furthermore,when(4.4.1)holds,wecanexpressdintermsofwasfollows:+'u0+w'u0+w216πN'2(u0+w)e±e−ed'λe=.(4.4.2)2e2(u0+w)Thetwopossiblechoiceofa“plus”or“minus”signin(4.4.2)isyetanotherindica-tionofmultipleexistencefor(4.1.4).Inaddition,forasolutionvof(4.1.4),itisnotdifficulttocheckthatifλed→+∞asλ→+∞,thendverifies(4.4.2)withthe(4.4.3)“plus”sign(forlargeλ).Thus,byvirtueofLemma4.4.20,toobtaina“non-topological-type”solution,itseemsreasonabletoimposethat(4.4.2)holdswiththe“minus”sign.Tothisend,inthespace(!)E:=w∈H():w=0,(4.4.4)weconsiderthesubsetAλ:={w∈E:wsatisfies(4.4.1)}.(4.4.5)Forλ>0sufficientlylarge,weseethat0∈Aλ,andsoAλisnotempty.Furthermore,foreveryw∈Aλwemaydefined±(w)bytheproperty+'u0+w'u0+w216πN'2(u0+w)e±e−ed±(w)'λe=,(4.4.6)2e2(u0+w)andthenconsiderthereducedfunctionals±(w)=JFλ(w+d±(w)),w∈Aλ.(4.4.7)λ±iscontinuousinANoticethatFλandcontinuouslyFrechetdifferentiablein´A˚λ,theλinteriorofthesetAλwhoseelementssatisfy(4.4.1)withthestrictinequality.Clearly,±,thenv=w+difw∈A˚λisacriticalpointforF±(w)isacriticalpointforJλandλ+orF−inA˚henceasolutionfor(4.1.4).SowemaysearchforcriticalpointsofFλ,λλinordertogetsolutionsfor(4.1.4),satisfying(4.4.2)respectivelywiththeassigned+or−sign.Towardsthisgoal,westarttopointoutthefollowinginterestingproperty(estab-lishedin[NT2])validfortheelementsofAλ. 1524PeriodicSelfdualChern–SimonsVorticesLemma4.4.20Ifw∈Aλandτ∈(0,1],then!1−τ!1λττeu0+w≤eτ(u0+w).(4.4.8)16πN1∈(0,1)sothatτa+2(1−a)=1.Forw∈AProof.Forτ∈(0,1],leta=λ,2−τwehave!!a!1−aeu0+w≤eτ(u0+w)e2(u0+w)!a!21−a≤eτ(u0+w)λeu0+w.16πNFromtheinequalityaboveand(4.4.1),theinequality(4.4.8)canbeeasilyderived.Onthebasisof(4.4.8),weobtainthefollowingresult:±areboundedfrombelowandcoerciveinAProposition4.4.21ThefunctionalsFλλwheretheyattaintheirinfimumatsomeelementw±∈Aλ.Namely,λF±(w±)=infF±.(4.4.9)λλλAλProof.Asadirectconsequenceof(4.4.2)weseethat!−1d+(w),ed−(w)≥4πNeu0+w,∀w∈Aeλ.λTherefore,!±(w)=J124πNu0+wFλλ(w+d±(w))≥||∇w||L2+4πNlog−4πNloge,2λandwecanuse(4.4.8)todeducetheestimate:1−τ!F±(w)≥1||∇w||2−4πNlogλeτ(u0+w)+4πNlog4πNλ2L2τ16πNλ!124πNτw≥||∇w||2−loge−cλ,τ2Lτforanyτ∈[0,1)andcλ,τasuitableconstant(dependingonlyonλandτ).AtthispointwecanapplytheMoser–Trudingerinequalityasstatedin(2.4.24),andfor0<τ0deriveN22±≥σ||∇w||2−cFλL2λ,τ,∀w∈Aλ. 4.4Analternativeapproach153Therefore,F±isboundedfrombelowandcoerciveinA±λλ.Hence,ifwnisamini-±inA±mizingsequenceforFλλ,thenwnisuniformlyboundedinH(),andwecanextractasubsequence(denotedinthesameway)suchthat±±u+w±u+w±pwn→wweaklyinH()ande0n→e0λinL(),∀p≥1.λInparticular,w∈Aλandd±(w±)→d±(w±).nλConsequently,infF±=limF±(w±)≥F±(w±)≥infF±,Aλλn→∞λnλλAλλand(4.4.9)isestablish.+belongstoA˚Actuallywecancheckthat,forlargeλ,theminimizerwλ,justbyλ+attainsintheboundaryofAcomparingtheminimalvaluethatFλ:λ!!!2.∂Aw∈H():w=0ande2(u0+w)=λeu0+w(4.4.10)λ=16πNwithitsvaluein0∈A˚λ(see[CY]and[Y1]fordetails).Thisfurnishesanalternativeprooftotheexistenceof“topological-type”solutionsfor(4.1.4),whenλissufficiently−belongstoA˚large.Onthecontrary,itisnotaseasytodeterminewhetherornotwλ.λSofar,thishasbeenverifiedin[T1]and[NT3],respectively,whenN=1andN=2.Toillustratethedifficultyoneencountersatthispoint,letµ=4πN.(4.4.11)Forw∈Aλ,evaluate:!−(w)=J12λu0+w+d−(w)2Fλλ(w+d−(w))=||∇w||L2+(e−1)+µd−(w)22⎛/0'⎞−112λ
µ04µe2(u0+w)=||∇w||L2+1−−µ⎝1+11−'2⎠22λλeu0+w⎛⎛/⎞⎞!0'u+w2µ04µe2(u0+w)+µ⎝−loge0+log−log⎝1+11−'⎠⎠λλeu0+w2λµ2µ=fλ(w)+−+µlog,22λwith⎛/⎞!0'12u+w04µe2(u0+w)fλ(w)=||∇w||−µloge0−µψ⎝1+11−⎠(4.4.12)L2'22λeu0+w 1544PeriodicSelfdualChern–SimonsVorticesand1ψ(ξ)=+logξ,ξ>0.(4.4.13)ξ−andfSincethefunctionalFλdifferonlybyaconstant,wealsoseethatfλisboundedλfrombelowinAλ,and−)=infffλ(wλ:=γλ.(4.4.14)λAλAsaconsequenceofthefactthatψin(4.4.13)isstrictlymonotoneincreasingforξ≥1,wefind:Lemma4.4.22Thefunctionλ−→γλ=inffλAλisstrictlymonotonedecreasing.Proof.For0<λ−1<λ2,observethatAλ⊂Aλ.Therefore,forwj=w,j=1,2,12λjwehave⎛/⎞!0'12u+w04µe2(u0+w1)γλ=fλ(w1)=||∇w1||2−µloge01−µψ⎝1+11−'2⎠112Lλ1u+we01⎛/⎞!0'12u+w04µe2(u0+w1)>||∇w1||2−µloge01−µψ⎝1+11−'2⎠2Lλ2eu0+w1=fλ(w1)≥fλ(w2)=γλ.222Expression(4.4.12)bringsourattentiontothefunctional!!I12u0+wµ(w)=|∇w|−µloge,w∈E(4.4.15)2whichwehavealreadydiscussed(overageneralsurface)inSection2.5ofChapter2.Infact,thefunctionalIµhasemergednaturallyfromourapproach,andweseethatitisboundedfrombelow,coerciveinAλ,andcontrolsthefunctionalfλfromaboveandbelowasfollows:Iµ(w)−µψ(2)≤fλ(w)≤Iµ(w)−µψ(1),∀w∈Aλ.Moreimportantly,thefollowingholds:Lemma4.4.23If−)=infffλ(wλE−∞.Hence,foreveryε>0,letwε∈EbesuchthatinfIµ≤Iµ(wε)0sufficientlylarge,weseethatwε∈Aλandwehave⎛/⎞0'−04µe2(u0+wε)infIµ−µψ(2)≤fλ(w)=inffλ≤Iλ(wε)−µψ⎝1+11−⎠λ'2EAλλeu0+wε≤infIµ+ε−µψ(2)+o(1)asλ→+∞.EInotherwords,inffλ−infIµ→−µψ(2)<−µψ(1),asλ→+∞.AλEHence,forλlarge(4.4.16)holds,andweconcludethefollowing:Corollary4.4.24Foreveryµ∈(0,8π]thereexistsλµ>0suchthat∀λ≥λµthe−satisfying(4.4.14)belongsinA˚extremalfunctionwλandsatisfies:λI−µ(w)−→infIµ,(4.4.18)λE'−1e2(u0+wλ)b→0,asλ→+∞.(4.4.19)λ:=
'2λu+w−e0λThus,settingw−andd−=d−(w−),λ=wλλλweseethatthefunction,v−2,λ=wλ+d,(4.4.20)λsatisfies− v=λeu0+v(1−eu0+v)−µ,||(4.4.21)v∈H(). 1564PeriodicSelfdualChern–SimonsVorticesInparticular,ifµ=4πNandN=1,2,(i.e.,thesingleanddoublevortexcase)thenv2,λin(4.4.20)solves(4.1.4).Sinced−2µλeλ=*',(4.4.22)'4µe2(u0+wλ)eu0+wλ1+1−'λ(eu0+wλ)2by(2.5.9)weseethatd−2µλeλ≤.(4.4.23)||HencebyLemma4.2.15(i),weknowthatthesolutionv2,λin(4.4.20)admitsthe“non-topological-type”behavior(4.2.31).Asamatteroffact,weshallseethatourconstructionallowsustostrenghten(4.2.31)asfollows:u+vu+w−+d−me02,λ=e0λλ→0asλ→+∞,inC()∩C(δ),(4.4.24)foranym∈Nandδ=∪NBδ(zj),δ>0small.j=1Toobtain(4.4.24),weneedtoestablishsomepreliminaryconvergencepropertiesforw−<0in,λ.Westartbyobservingthattheproperty,u0+v2,λ=u0+wλ+dλand(4.4.22)implythat!ueu0+wλ0+wλ−log≤logλ−logµ.(4.4.25)Inaddition,bystraightforwardcalculationswecheckthat
eu0+wλ1− wλ=µ'eu0+wλ−||+fλ,µ,'(4.4.26)wλ∈H():wλ=0witheu0+wλe2(u0+wλ)fλ,µ=aλ(µ)'−',(4.4.27)eu0+wλe2(u0+wλ)and'4µ2e2(u0+wλ)a.(4.4.28)λ(µ)=*'2'24µe2(u0+wλ)λe2(u0+wλ)1+1−'λ(eu0+wλ)2By(4.4.19),wehavethataλ(µ)=2µ2bλ(1+o(1))→0,andso||fλ,µ||L1()→0asλ→+∞.Wealsoderive||fλ,µlog|fλ,µ|||L1()≤Cµbλlogλ,(4.4.29)withCµ>0asuitableconstantindependentofλ.Theestimateabovemotivatesthefollowing: 4.4Analternativeapproach157Lemma4.4.25Forbλin(4.4.19),wehave:liminfbλlogλ=0.(4.4.30)λ→+∞Proof.Weknowthatthereexistsalargeλ∗>0,suchthatforλ≥λ∗thefunctionγλin(4.4.14)ismonotonedecreasingin[λ∗,+∞),andthereforedifferentiablefora.e.λ≥λ∗.Furthermore,!λγds=γ−λ−γλ−→γλ−infIµ+µψ(2):=C,asλ→+∞.(4.4.31)s∗∗λ∗EOntheotherhand,forδ>0wehave111(γλ−γλ+δ)=(fλ(wλ)−fλ+δ(wλ+δ))≥(fλ(wλ)−fλ+δ(wλ))δδδ⎛⎛/⎞0'µ04µe2(u0+wλ)=−⎝ψ⎝1+11−⎠'2δλeu0+wλ⎛/⎞⎞0'04µe2(u0+wλ)−ψ⎝1+11−⎠⎠,'2λ+δeu0+wλandpassingtothelimitasδ→0+,fora.e.λ>λ∗,wefind:⎛/⎞0''04µe2(u0+wλ)2µ1e2(u0+wλ)−γ≥µψ⎝1+11−⎠*λλ'u+wλ2λ2'e2(u0+wλ)'u+wλ2e04µe01−λ'u0+wλ2(e)1=aλ.2λTherefore,bytheintergrabilitypropertyof−γin[λ∗,+∞)(see(4.4.31)),wededuceλthatliminfaλlogλ=0.λ→+∞Sinceaλ(µ)=2µ2bλ(1+o(1))asλ→+∞,weconclude(4.4.30).Bymeansoftheblow-upanalysisofthefollowingchapter,weshallbeabletoreplacethe“liminf”with“lim”in(4.4.30)(seeProposition6.4.14).Westarttoanalyzetheeasiercase,µ∈(0,8π),wherethefollowingstrongerstatementholds:Proposition4.4.26Ifµ∈(0,8π)andm∈Z+,thenthereexistsaconstantC>0(dependingonmandµonly)suchthatforv−,wehave2,λ=wλ+dλ||eu0+wλ||+λ||eu0+v2,λ||≤C,(4.4.32)Cm()Cm() 1584PeriodicSelfdualChern–SimonsVorticesforλ>0sufficientlylarge.Inparticular,u+vu+w−||e0λ,2||m=||e0λ+dλ||→0,asλ→+∞.(4.4.33)C()Cm()Proof.Sinceforµ∈(0,8π)thefunctionalIµiscoerciveinEand(4.4.18)holds,thenitfollowsthatwλisboundedinEuniformlyinλ.Consequently,theMoser–Trudingerinequality(2.4.24),appliedtogetherwithJensen’sinequality(2.5.8),impliesthate2(u0+wλ)eu0+wλ'+'≤C,foreveryp≥1,e2(u0+wλ)eu0+wλpLp()L()withaconstantC>0independentofλ.Hence,theright-handsideoftheequationin(4.4.26)isuniformlyboundedwithrespecttoλinLp()-norm,∀p≥1.Thus,byellipticestimatesandabootstrapargument,wegetthatwλisboundedinCm(),uniformlyinλ,∀m∈Z+.Consequently,forasuitableconstantC>0,wefindu+w−−||e0λ+dλ(wλ)||≤Cedλ,Cm()and(4.4.32)and(4.4.33)readilyfollowfrom(4.4.23).Fromtheargumentabove,weseethatwhenµ∈(0,8π),thenalongasequenceλn→+∞,wλ→winCm(),withwaminimizerforIµinE.Hence,wsatisfiesnthe“limiting”meanfieldequationoftheLiouville-type:⎧⎪⎪eu0+w1⎨− w=µ'−,eu0+w||!(4.4.34)⎪⎪⎩w∈H()andw=0,forµ∈(0,8π).Atthispoint,itwouldbeparticularlyusefultoknowifuniquenessholdsforthesolutionof(4.4.34)orevenfortheminimizerofIµ.Infact,thisinformationwouldimplyconvergenceforthewholefamilywλtosuchsolution(orminimizer)asλ→+∞.However,asdiscussedinSection2.5ofChapter2,thequestionofuniquenessfor(4.4.34)posesaquitedelicateproblem,yetnotresolved,inspiteoftherecentcontributionscontainedin[CLS],[LiL],[LiL1],[LiW],and[LiW1].Forinstance,itisnaturaltoexpectthattheexplicitsolutionofOlesenin[Ol](obtainedbyasuitableuseoftheWeierstrassP−functionintheLiouvilleformulaof(2.2.3))shouldbetheuniquesolutionfor(4.4.34)whenµ=4π.Whenµ=8π,itismuchhardertocarryoutadetailedasymptoticanalysisofwλasλ→+∞.Suchanalysisrelatestotheextremalpropertyofthe(nolongercoercive)functionalIµ=8πasfollows:Lemma4.4.27Letµ=8π,thenw−λ(=w)in(4.4.14)isboundedinE,anduni-λformlysoinλ,ifandonlyifIµ=8πattainsitsinfimuminE. 4.4Analternativeapproach159Proof.Clearly,ifwλisuniformlyboundedinE,thenasabove,wefindthatalongasubsequenceλn→+∞,wλconvergesinE(andinanyotherstrongerCm()-norm)ntoaminimizerforIµ=8π.Viceversa,supposethatthereexistsw0∈E,suchthatIµ=8π(w0)=infIµ=8π.EThenforλ>0sufficientlylarge,wehavew0∈Aλand⎛/⎞0'032πe2(u0+w0)Iµ=8π(w0)−8πψ⎝1+11−⎠=f'2λ(w0)λeu0+w0⎛/⎞0'032πe2(u0+wλ)≥fλ(wλ)=Iµ=8π(wλ)−8πψ⎝1+11−⎠'2λeu0+wλ⎛/⎞0'032πe2(u0+wλ)≥Iµ=8π(w0)−8πψ⎝1+11−⎠.'2λeu0+wλConsequently,⎛/⎞⎛/⎞0'0'032πe2(u0+wλ)032πe2(u0+w0)ψ⎝1+11−'2⎠≥ψ⎝1+11−'2⎠λeu0+wλλeu0+w0andusingthefactthatψismonotonedecreasingin[1,+∞),weconcludethat''e2(u0+wλ)e2(u0+w0)≤,forλ>0sufficienltylarge.(4.4.35)'2'2eu0+wλeu0+w0Hence,recalling(4.4.26),wedecomposewλ=w1,λ+w2,λ,(4.4.36)withw1,λtheuniquesolutionfortheproblem⎧⎪⎪eu0+wλ1⎨− w1,λ=8π'−,in,eu0+wλ||!(4.4.37)⎪⎪⎩w1,λ∈H()andw1,λ=0,andw2,λtheuniquesolutionfortheproblem⎧⎨− w2,λ=fλin,!(4.4.38)⎩w2,λ∈H()andw2,λ=0,wherefλisdefinedby(4.4.27)–(4.4.28)withµ=8π. 1604PeriodicSelfdualChern–SimonsVorticesFrom(4.4.35),weseethattheright-handsideoftheequation(4.4.37)isuniformlyboundedinL2()-norm,andhence||w1,λ||C0()≤C1,(4.4.39)∀λ>0,withasuitableconstantC1>0.While,byGreen’srepresentationformula(cf.[Au]),wehave!!11w2,λ(z)=logfλ(y)dy+γ(z−y)fλ(y)dy,(4.4.40)2π|z−y|withγasmoothdoublyperiodicfunction.Hence,bymeansofJensen’sinequality(2.5.6),wefind!2π|w2,λ|!!||fλ||1|fλ(y)|eL1()dx≤dxdy+C1|x−y|||fλ||L1()!!|fλ(y)|1≤dxdy+C2≤C,||fλ||L1()|x−y|≤1|x−y|forsuitablepositiveconstants
C1,C2,andC.Since,by(4.4.35)weknowthat||f1λ||L1()=Oλ,wecanusetheestimatesaboveincombinationwith(4.4.39)toassertthatu0+wλ∀p>1∃λp>0andCp>0suchthat||e||Lp()≤Cp,∀λ≥λp.Inotherwords,theright-handsideoftheequationin(4.4.26)isuniformlyboundedinLp()foreveryp>1,andthisimpliesthatwλisuniformlyboundedinEasclaimed.Lemma4.4.27bringstoourattentiontheexistenceofaminimizerfortheextremalproblem:infIµ=8π.(4.4.41)EInfact,whentheinfimumin(4.4.41)isattained,thentheconclusionofProposition4.4.26holdsalsowhenweletµ=8π.Remark4.4.28Wewishtoemphasizethatforµ=8π,thefamilywλconstructedaboveprovidesuswithaparticularly“nice”minimizingsequenceforthefunctionalIµ=8π.Moreover,theconstructionpresentedaboveworksequallywellforthefunctionalin(2.5.1),whereweconsideramoregeneralclosedsurfaceMandassumethat(2.5.3)holds.InSection6.3ofChapter6weshalltakeadvantageofproperties(4.4.26)–(4.4.30)inordertoprovethat,forM=R2
aZ×ibZanda,b>0,theinfimumofIµ=8πisattainedinEprovidedthatthereexistsp∈M:u0(p)=maxMu0>0,andu8π>0.0(p)+|M| 4.4Analternativeapproach161Unfortunatelysuchaconditionisjustmissedifwetakeu0definedby(4.1.3)andN=2.Interestinglyinthiscase,Chen–Lin–Wangin[ChLW]haveprovedthattheinfimumin(4.4.41)isneverattained,whenin(4.1.3),wetakez1=z2(i.e.,asinglevortexwithdoublemultiplicity).Evenmorestrongly,recentlyin[LiW]ithasbeenproventhatthecorrespondingEuler–Lagrangeequation(4.4.34)(withµ=8π)ad-mitsnosolutions.Seealso[LiW1]formorerecentdevelopmentsinthisdirection.Thus,returningtoourdoublevortexproblem,wemustbereadytoanalyzewλunderthecircumstancethat,asλ→+∞,!||∇wuo+wλλ||L2()→+∞orequivalentlymaxwλ−loge→+∞.(4.4.42)Tothisend,weshallneedmoresophisticatedanalyticaltoolsinvolvingblow-uptech-niques,thatwewilldevelopinthefollowingchapterinordertoconcludethefollowingresult:Theorem4.4.29Let:={z=at+isb}forsomea,b>0.Forµ=8π,thefunc-−(w−−−−tionv2,λ=wλ+dλ=wandd=d(w))definesasolutionof(4.1.4)withλλλλu0givenin(4.1.3)andN=2.Itsatisfies||eu0+v2,λ||→0,asλ→+∞.(4.4.43)C0()Furthermore:(i)Either(4.4.41)isattainedatsomew0∈E,andalongasequenceλn→+∞,wehaveeu0+w0λneu0+v2,λn(1−eu0+v2,λn)→8π'inL1();(4.4.44)eu0+w0(ii)or(4.4.41)isnotattained,andalongasequenceλn→+∞,wehaveλneu0+v2,λn(1−eu0+v2,λn)→8πδpweaklyinthesenseofmeasurein,(4.4.45)0andu0(p0)=maxu0.Remark4.4.30Observethatbythenon-existenceofextremalsforIµ=8πderivedin[ChLW],weknowthatalternative(ii)alwaysholdswhenweconsiderasinglevortexwithdoublemultiplicity(i.e.,z1=z2in(4.1.3)).TocompletetheproofofTheorem4.4.29,weneedtoshowthatwhen(4.4.42)holds,then(4.4.43)and(4.4.45)aresatisfied.ThiswillbethegoalofSection6.4inChapter6.Forthemoment,wewishtopointoutthatbymeansof(4.4.29),wecanuse(4.4.40)toprovidethefollowingestimateforw2,λin(4.4.38):2+max|w||∇w2,λ||22,λ|=O(bλlogλ),asλ→+∞.L() 1624PeriodicSelfdualChern–SimonsVorticesTherefore,alongasequenceλn→+∞suchthatbnlogλn→0(alwaysavailablebyLemma4.4.25),forwn=wλ−w2,λ,nnwefind
'hnewn1− wn=8πhnewn−||in,'(4.4.46)wn∈H(),wn=0,withhn=eu0+w2,nand!12wn||∇wn||2−8πloghne→infIµ=8π.(4.4.47)2L()ENoticeinparticularthatforhnwehave$N|z−z2hn(z)=j|Vn(z)andVn→V>0uniformlyin.(4.4.48)j=1Weshalldevotethenextchaptertotheasymptoticanalysisofsequencesforwhich(4.4.46)and(4.4.48)hold.4.5MultipleperiodicChern–SimonsvorticesWecansummarizetheresultsestablishedaboveinthefollowingstatementconcerningperiodicChern–Simonsvortices:Theorem4.5.31ForagivenN∈NandasetofpointsZ:={z1,...,zN}⊂(not+||necessarilydistinct),thereexistsaconstant0max{2N1+N2,2N2+N1}.(4.5.18)3||Moreimportantly,in[NT1]itisshownthatforapair(w1,w2)∈E×Esatisfying'h2e2wa3λa'2≤,a=b∈{1,2},(4.5.19)haewa32π(2Na+Nb)itispossibletoobtainfourpairsofsolutionsfor(4.5.17),thatdependonw1andw2andthatarespecifiedaccordingtothepossiblechoicesofthe“+”or“−”signinthefollowingequivalentformulationofthesystem(4.5.17):''haewa+edbh1h2ew1+w2eda='4h2e2waa+'wad'w+w232π'22wahae+ebh1h2e12−(2Na+Nb)hae3λ±',4h2e2waa(4.5.20)wherea=b∈{1,2}.Thus,wedenotewith(d+,d+)and(d−,d−)thesolutionsfor(4.5.20)wherewe1212pickthe“+”signorthe“−”sign,respectively,inbothequationsin(4.5.20).Here(d±,d±)(respectively(d∓,d∓))denotesthesolutionforwhichwepick:the“+”sign1212(respectively,the“−”sign),intheequationcorrespondingtoa=1in(4.5.20);andthe“−”sign(respectively,the“+”sign),intheequationcorrespondingtoa=2in(4.5.20).Wereferto[NT1]forthedetailedproofontheexistenceofsuchsolutions.Therefore,asfortheabeliancase,onthesetAλ={(w1,w2)∈E×E:(4.5.18)holds}⊂E×E,(4.5.21)wedefinefourconstraintfunctionals:F∗(w∗∗λ1,w2)=Jλ(w1+d1,w2+d2)with∗=+or−(4.5.22)andF#(w##λ1,w2)=Jλ(w1+d1,w2+d2)with#=±or∓.(4.5.23)Since(4.5.18)hasthesamestructureas(4.4.1)(infact,reducestoitwhenN1=N2),itisnotdifficulttocheckthataresultsimilartoLemma4.4.20remainsvalidforua0+wa=hwa−1ijeae,a=1,2.Furthermore,intermsoftheK=(K)i,j=1,2,the2−1inverseoftheCartan2×2matrix,K=,wehave−122!1ijJλ(w1+d1,w2+d2)=K∇wi·∇wj+λW(w1+d1,w2+d2)2i,j=14π+(2N1+N2)d1+(2N2+N1)d2,(4.5.24)3 1684PeriodicSelfdualChern–SimonsVorticeswithWgivenin(4.5.14).AsinProposition4.4.21thefunctionalsin(4.5.22)and(4.5.23)areboundedfrombelowandcoerciveinAλ,wheretheyattaintheirinfimum.See[NT1]fordetails.
Hence,weletw∗,w∗∈Aλ1,λ2,λF∗w∗,w∗=infF∗,(4.5.25)λ1,λ2,λλAλwith∗=+
or−;andforw#,w#∈Aλ,1,λ2,λ
F#w#,w#=infF#,(4.5.26)λ1,λ2,λλAλwith#=±or∓.WeneedtocheckthattheseextremalfunctionsbelongtotheinteriorofAλ,inordertoyieldasolutionfor(4.5.11)asgivenby:v∗=w∗+d∗,a=1,2,(4.5.27)a,λa,λawith∗=+or−;andv#=w#+d#,a=1,2,(4.5.28)a,λa,λawith#=±or∓.For∗=+,weexpect(4.5.27)tosatisfythetopologicalasymptoticbehavioru0+v+eaa,λ→1,a=1,2;asλ→+∞.(4.5.29)Whilefor∗=−,weexpect(4.5.28)tosatisfythenon-topologicalasymptoticbehavioru0+v−eaa,λ→0,a=1,2;asλ→+∞.(4.5.30)Similarly,for#=±or#=∓,weexpect(4.5.28)tosatisfyrespectivelyoneofthefollowingmixedtopological/non-topologicalconditionu0+v±1u0+v±e11,λ→ande22,λ→0,(4.5.31)2oru0+v∓u0+v∓1e11,λ→0ande22,λ→;(4.5.32)2asλ→+∞.Asfortheabeliancase,thistaskcanbecarriedoutmoreeasilyforthetopologicalcase,correspondingtothechoice∗=+,andleadstothefollowingresult: 4.5MultipleperiodicChern–Simonsvortices169Theorem4.5.32[NT1]Forsufficientlylargeλ,wehave:
infF+>minF+=F+w+,w+;λλλ1,λ2,λ∂AλAλandtherefore(v+=w++d+)a,λa,λaa=1,2definesasolutionfor(4.5.11)andsatisfies(4.5.29)inLp(),p≥1,andpointwisea.e.in.Actually,(v+,v+)definesa1,λ2,λlocalminimumforthefunctionalJλin(4.5.13).FurthermoreJλadmitsalsoanothercriticalpointinE×E.Thus,asfortheabeliancase,wecanalwaysensuremultiplesolutionsfor(4.5.11),andalsoguaranteethatonesuchsolutionalwaysadmitsatopologicalbehavior,inthesensethat(4.5.29)holds,asλ→+∞.Wereferto[NT1]fordetails.Onthecontrary,theothercasesaremoredelicatetohandleandrequirefurtherrestrictionsonthevortexnumbersNa,a=1,2.Concerningthechoiceof∗=−,whichweexpecttoyieldthenon-topologicalbehavior(4.5.29),thecondition,
infF−>minF−=F−w−,w−,(4.5.33)λλλ1,λ2,λ∂AλAλwascheckedtoholdbyNolasco–Tarantelloin[NT1]undertheassumptionthatN1+N−,w−)alsosatisfiesastrongconvergence2=1.Inthissituation,theminimizer(w1,λ2,λproperty(alongasequenceλn→+∞)towardsaminimizeroftheTodafunctional:1!22!(wijwjIµ1,µ21,w2)=K∇wi·∇wj−µiloghje,w1,w2∈E,2i,j=1j=1(4.5.34)with4π4πµ1=(2N1+N2)andµ2=(2N2+N1).(4.5.35)33Thiscanbeexplainedbyrecallingexpression(4.5.24)forJλandthataccordingtoJost–Wang’sresultin[JoW1](seeTheorem1.2.13),thefunctionalin(4.5.34)isboundedfrombelowifandonlyifµj≤4π.Moreover,theinfimumisalwaysat-tainedwhenthestrictinequalityholds,i.e.,whenµj<4π,∀j=1,2.ClearlyifN1+N2=1,thenµ1andµ2in(4.5.35)arebothlessthen4π;thisjustifiestheneedforsuchrestrictiononthevortexnumbersN1andN2.Ontheotherhand,thefunctionalin(4.5.34)remainsboundedfrombelowalsowhenµ1andµ2aregivenby(4.5.35)withN1=N2=1.Inthiscase,µ1=µ2=4πandweareataborderlinesituation,analogoustothatencounteredforthedoublevortexcase(i.e.,N=2)intheabelianChern–Simons6th-ordermodel.Indeed,onecanproceedinananalogousway(see[NT1])toarriveatthefollowingconclusion: 1704PeriodicSelfdualChern–SimonsVorticesTheorem4.5.33([NT1])IfNa∈{0,1},a=1,2andλ>0issufficientlylarge,−,w−)ofF−belongstotheinteriorofthesetAthentheminimizer(wλin(4.5.21).1,λ2,λλ−=(w−+d−)Henceva,λa,λaa=1,2definesasolutionfor(4.5.11)thatadmitsthefollowing“non-topological-type”asymptoticbehavior,u0+v−eaa,λ→0uniformlyinasλ→+∞,a=1,2.(4.5.36)Fordetailswereferto[NT1]forthecaseN1+N2=1,forwhichtheconvergencein(4.5.36)isactuallyshowntoholdwithrespecttoCm()-norm.Inaddition,alonga−→wsequenceλn→+∞,itisshownthatwainEwitha=1,2,and(w1,w2)aa,λnsolutionofthe2×2Todasysteminmeanfieldform(recall(2.5.25)):⎧⎪⎪4π2h1ew1h2ew24πN1⎪⎪− w1=(2N1+N2)'w−(2N2+N1)'w−in,⎪⎪3h1e1h2e2||,⎨4π2h2ew2h1ew14πN2− w2=(2N2+N1)'−(2N1+N2)'−in,(4.5.37)⎪⎪3h2ew2h1ew1||⎪⎪!!⎪⎪⎩w1,w2∈H(),w1=0=w2.Moreprecisely,weknowthat(w1,w2)definesaminimumforthefunctionalin(4.5.34)andin(4.5.35).Themoredelicatecase,N1=N2=1,followsbyargumentssimilartothosegivenabovefortheabelianChern–Simons2-vortexproblem,incombinationwiththeresultsin[JoW1]and[JoW2].Analogously,inthissituation,onecannolongerguaranteestrongconvergencetowardsasolutionofthesystem(4.5.37).Infact,forN1=N2=1,theexistenceofasolutionfor(4.5.37)becomesadelicateissuethatingeneralcannotbedealtbyusingasimpleminimizationprocedure.Apossibilitywouldbetosupposethath1andh2satisfyconditionsanalogousto(2.5.4)(asshownin[JoLW]towhichwereferfordetails);howeverthisassumption,doesnotallowustotreatourchoiceofhain(4.5.8)and(4.5.10)fora=1,2.Finally,concerningtheminimizationproblemin(4.5.26),theconditionN1+N2=1isstillsufficienttoguaranteethateitheroneoftheminimizers,(w±,w±)forF±or1,λ2,λλ∓,w∓)forF∓,belongstotheinteriorofthesetA(wλ(in(4.5.21)),whenλis1,λ2,λλsufficientlylarge.Morepreciselythefollowingholds:Theorem4.5.34([NT1])IfN1=0andN2=1,thenforλ>0sufficientlylargethe±,w±)ofF±belongstotheinteriorofA±±minimizer(wλ,sothat(v)=(w+1,λ2,λλa,λa,λ±)da=1,2definesasolutionfor(4.5.11).Furthermore,a,λw±→0,asλ→+∞,andw±isuniformlyboundedinE;1,λ2,λ±1±d→logandd→−∞asλ→+∞122Inparticular,v±1u0+v±e1,λ→ande22,λ→0,asλ→+∞.2(NoticethatifN1=0,thenu0≡0.)1 4.6Finalremarksandopenproblems171IfN1=1andN2=0,thentheconclusionaboveholdsfortheminimizer(w∓,w∓)ofF∓,withtheroleoftheindexa=1anda=2exchanged.1,λ2,λλIfN±∓1=0andN2=1,(orifN1=1andN2=0),thenw(orw)isuniformly2,λ1,λboundedinE,andalongasequenceλ±(orw∓)convergesn→+∞,wefindw2,λn2,λn(weakly)inEtoasolutionofthemeanfieldequation(4.4.34)withµ=4π.IntermsofperiodicChern–SimonsSU(3)-vortices,theresultsabovecanbesum-marizeasfollows:Theorem4.5.35Fora=1,2,letNa∈Z+andZa=za,...,za⊂bea1Nagivensetofpointsrepeatedaccordingtotheirmultiplicity.Thereexists0,NOsuchsolutionsexists.8πmax{2N1+N2,2N2+N1}Asfortheabeliancase,itisreasonabletoexpectthatconclusions(iii)and(iv)ofTheorem4.5.35shouldholdwithoutanyrestrictiononthevortexnumbersNa,a=1,2.4.6FinalremarksandopenproblemsAgainweconcludeourdiscussionofperiodicChern–Simonsvorticesbypointingoutsomeoftherelatedopenproblems.Firstly,thediscussionaboveexplainsandre-enforcesourinterestintheexistenceornon-existenceofextremalsfortheMoser–Trudingerinequality,aquestiontowhichweprovideapartialanswerinthefollowingchapter. 1724PeriodicSelfdualChern–SimonsVorticesThisquestionnowholdsevenmorerelevanceinthecontextofsystems,andmorespecifically,fortheSU(3)-Todafunctional(4.5.34),whereweask:Openquestion:Doesthefunctional(4.5.34)withNa=1andhadefinedin(4.5.8)and(4.5.10)witha=1,2,attainitsinfimuminE×E?Furthermore,howdoestheanswertothisquestiondependontheshapeoftheperiodiccelldomain(i.e.,rectangular,rhombus,etc.)?Also,forarectangulardomain,doestheconditionthatthepointsza,a=1,21(in4.5.8),coincideornotaffecttheanswer?Inthisrespect,recalltheresultsin[LiW]forthesingleequation.Clearly,theexistenceofextremalscanbemoregenerallyaskedfortheSU(n+1)-Todafunctionaloveraclosedsurface.Asusual,weexpectthecaseofthestandardsphereM=S2tocontainmanyelementsofinterest.Wementionthattheonlyavail-ableresultinthiscontextiscontainedin[JoLW].Ananswertotheabovequestionswouldpermitustoconstructnon-abelianChern–Simonsvorticessatisfyingappropriate“concentration”andstrong“localization”prop-ertiesaroundsomepoints,inamannerconsistentwithwhatisobservedinthephysicalapplications.Asamatteroffact,evenintheabeliancaseandforanyvortexnumberN≥2,itisnotknownwhetherperiodicChern–Simonsvorticesexistwhichasymptoticallysatisfythe“non-topological-type”condition(4.5.2),andatthesametime,admita“concentration”behaviorsimilartothatin(4.5.1).AnyprogressinthisdirectionwouldcertainlygiveindicationsonhowtocarryoutsimilarconstructionsalsoforSU(3)-vortices,possiblywithrespecttoboth“non-topological,”and“mixed-type”asymptoticbehaviors,asgivenin(4.5.6)and(4.5.7),respectively.Inanyevent,removingtherestrictiononthevortexnumbersNa,a=1,2,fromTheorem4.3.35wouldserveasanimportant,steptowardtheunderstandingofSU(3)-vortices.Thisamountstoprovideanaffirmativeanswertothefollowing:Openquestion:ConsideranygivenNa∈Nandhasatisfying(4.5.8),(4.5.10),a=1,2.Doesproblem(4.5.11)admitasolution(v1,v2)λforlargeλ>0suchthatforλ→+∞,hva→0a.e.in,a=1,2ae(non-topological-type);andasecondsolution(v˜1,v˜2)λ,forλ>0large,suchthatasλ→+∞,h1ev1→1andh2ev2→0orh1ev1→0andh2ev2→122a.ein(mixed-type)? 5TheAnalysisofLiouville-TypeEquationsWithSingularSources5.1IntroductionTheconstructionofperiodic“non-topological-type”Chern–Simonsvorticesdevel-opedinSection4.4ofChapter4hasleadustoconsidersequenceswkdefinedoverthe(periodiccell)domainsatisfying⎧w⎪⎪hkek1⎨− wk=µk'w−in,hkek||!(5.1.1)⎪⎪⎩wk∈H():wk=0,whereµk>0isagivenboundedsequenceandtheweightfunctionhktakestheform$mh2αjk(z)=|z−pj|Vk(z)and00asthecorrespondingmultiplicity.Afteratranslation,wecanalwaystakethevortexpointtocoincidewiththeorigin.Moreover,ifweworkwiththenewsequence!hwkuk=wk−logke, 1745TheAnalysisofLiouville-TypeEquationsWithSingularSourcesthenweareleadtostudythefollowing“local”problem:⎧2αVuk⎨−!uk=|z|k(z)einBδ(0)={|z|<δ},|z|2αVuk(5.1.4)⎩k(z)e≤C,Bδ(0)whereδ>0small,C>0isasuitableconstant,and00,thentheformationof“multiplebubbles”isnotpossible,whilesuchphenomenonisknowntooccurwhen(5.1.5)isviolated(seeexamplesinSection5.5.5).Noticethatcondition(5.1.5)canalwaysbecheckedforouroriginalsequencewkin(5.1.1).Inthe“single-bubble”situation,weshallbeabletoprovidepointwiseestimatesontheprofileofthesequenceukaroundits“bubblingpoint”.Thisanalysisisbaseduponanappropriate“inf+sup”estimateforsolutionsoftheequation2αV(z)euinB−u=|z|1(0),(5.1.6)whichholdsindependentinterest.Moreprecisely,itcanbeshownthat,ifα≥0andV∈C0,1(B1)satisfies00dependingonlyα,b1,b2,andA.Afterour“local”analysisiscompleted,weshallpatchtogetherallsuch“local”informationtoobtaina“global”concentration/compactnessprincipleforasolution-sequencesatisfying(5.1.1),(5.1.2)(seeTheorem5.4.34andTheorem5.7.61below).BysuchresultsinChapter6,weshallbeabletocompletetheproofofTheorem4.4.29andalsoobtainanexistenceresultforageneralizationofthemeanfieldequation(2.5.2)whichisusefulinthestudyofperiodicelectroweakvortices.Thematerialofthischapterfollowsclosely[T4]andcollectsworkof[BM],[BLS],[LS],[L2],[BT1],[BT2],[BCLT],[T5],and[T6]. 5.2Backgroundmaterial1755.2BackgroundmaterialInthissectionwecollectsomebasicpropertiesconcerningsolutionsoftheLiouville-typeequation−u=Weuin,(5.2.1)where⊂R2isaboundedopenregulardomainandWagivenfunction.Mostoftheresultsherearestated,keepinginmindthefollowingmodelweightfunction:W(z)=|z|2αV(z),α∈R+,V∈C()and0∈.(5.2.2)WestartbyderivinganinequalityoftheJohn–Nirenbergtypeandweuseittoshowthatifu∈L1()satisfies(insenseofdistributions)loc−u=fin,(5.2.3)andf∈L1(),thenloc|u|pe∈L(),p≥1.(5.2.4)loc1,qNoticethatbymeansofellipticregularitytheory,wegeteasilythatu∈W()forloc1≤q<2.Sincethepowerq=2isjustmissed,(5.2.4)cannotbededucedsimplyby(localizing)theMoser–Trudingerinequality.Lemma5.2.1Letf∈L1()andu∈W1,q(),11andu∈L()besuchthate∈loclocp11Lloc()withp+p=1.Ifusatisfies(5.2.1)(inthesenseofdistributions),then2,pβ2,βu∈W().Ifinaddition,W∈C()β∈(0,1],thenu∈C()anditlocloclocdefinesaclassicalsolutionfor(5.2.1).Proof.Letf=Weu.Bythegivenassumptionsf∈L1()andthereforewecanlocuseCorollary5.2.4toseethate|u|∈Lp(),∀p≥1.Consequently,f∈Ls()loclocforsomes>1,andthereforewecanuseellipticregularitytheorytofindthatu∈2,s∞pW()forsomes>1.Inparticular,u∈L()andthereforef∈L().Inloclocloc2,pββturn,u∈W()asclaimed.IfwealsohavethatW∈C(),thenf∈C()locloclocandthedesiredconclusionfollowsbySchauder’sestimates(cf.[GT]).1,2Corollary5.2.6Letu∈W()satisfy(5.2.1).Wehave:locp2,pifW∈L()forsomep>1,thenu∈W();loclocβ2,βifW∈C()forsomeβ∈(0,1],thenu∈C()andudefinesaclassicalloclocsolutionfor(5.2.1).Proof.Simplyobservethatinthiscaseby(localizing)theMoser–Trudingerinequality|u|p(2.4.9),weknowthate∈L(),∀p≥1.Therefore,Lemma5.2.5appliestoulocandyieldstothedesiredconclusion.Remark5.2.7Corollary5.2.6canbeusedtojustifytheregularityofthevariousweaksolutionsconstructedinChapters3,4,and7fortheellipticproblemsarisinginChern–Simonsandelectroweakvortextheory.Similarly,thisallowsustosolvethemeanfieldequationweaklyintheSobolevspaceH1bymeansofvariationalmethods.Next,wewishtopointoutaHarnack-typeinequality,validforsolutionsof(5.2.1)whenWsatisfies(5.2.2).Tothispurpose,westartwiththefollowing:Proposition5.2.8Letf∈Lp()forsome10,dependingon||andponly,suchthatsupu≤βinfu+(1+β)γ#f#Lp().(5.2.8)Proof.Inequality(5.2.8)isjustadirectconsequenceofHarnack’sinequality.Tothispurpose,letwbetheuniquesolutionfortheDirichletproblem:− w=fin,w=0in∂.Sincef∈Lp()andp>1,standardellipticestimates(see[GT])implythatmax|w|≤γ#f#Lp(),(5.2.9)¯ 1785TheAnalysisofLiouville-TypeEquationsWithSingularSourceswithγ>0asuitableconstantdependingon||andponly.Moreover,weseethatthefunctionw−udefinesanharmonicfunctioninwhichisnonnegativein∂.Sow−uisnonnegativein,andwecanapplyHarnack’sinequalitytoobtainaconstantβ∈(0,1),dependingonandonly,suchthat1sup(w−u)≤inf(w−u)(5.2.10)βFrom(5.2.9)and(5.2.10),wederivesupu≤βinfu+(1+β)max|w|≤βinfu+(1+β)γ#f#Lp(),asclaimed.Forlateruse,wepresentthefollowingconsequenceofProposition5.2.8.Corollary5.2.9Thereexistsa(universal)constantβ∈(0,1)suchthat,ifξsatisfies− ξ=ginB2R,ξ≤Cin∂B2R,withg∈Lp(B2R)forsome10dependingonponly.Proof.Letu(z)=ξ(Rz)−C.Itsatisfies(5.2.5)in=B2withf(z)=R2g(Rz)∈Lp(B2).So,wecanapplyProposition5.2.8touwith=B1toobtainauniversalconstantβ∈(0,1)andγp>0dependingonponly,suchthatsupu≤βinfu+(1+β)γp#f#Lp(B2).B1B1Fromtheaboveinequality,weeasilyderivethedesiredconclusion.
Proposition5.2.10Thereexistsa(universal)constantβ∈(0,1)suchthatforagivenb>0,α≥0,andC>0,everysolutionuof(5.2.1)in:=r≤|z|≤2R,with22αV(z),#V#W(z)=|z|L∞≤bandsup{u(z)+2(α+1)log|z|}≤C,(5.2.11)satisfiessupu≤βinfu+2(α+1)(β−1)logρ+c,(5.2.12)|z|=ρ|z|=ρforeveryρ∈(r,R)andasuitableconstantc>0dependingonlyonα,b,andC. 5.2Backgroundmaterial179Remark5.2.11WewishtostressoncemorethatneitherβorcdependonrandR.Furthermore,property(5.2.11)willappearasanaturalconditioninthesequel.ProofofProposition5.2.10.Foragivenρ∈(r,R),letv(z)=u(ρz)+2(α+1)logρ,(5.2.13)satisfying:()2αv1− v=|z|V(ρz)einD:=<|z|<2.2Thus,settingf(z)=|z|2αV(ρz)evandrecalling(5.2.11)weseethatsupv≤C+2(α+1)log2:=C1D¯andC#f#L∞(D)≤4be.Therefore,wecanapplyProposition5.2.8tov−C1in={z:|z|=1}⊂Dtoobtain(universal)constantsβ∈(0,1)andγ>0,suchthatsupv≤βinfv+(1+β)γ#f#L∞(D)+(1−β)C1.|z|=1|z|=1Fromtheinequalityaboveandbymeansof(5.2.13)weimmediatelyderive(5.2.12).WeconcludethissectionwithausefulPohozaev-typeidentityvalidfor(smooth)solutionsof(5.2.1):Pohozaev’sidentity:LetW∈W1,∞()andu∈C2()satisfy(5.2.1).Thefol-lowingidentityholdsforeveryregularsubdomainD⊆:!!|∇u|2z·ν−(ν·∇u)(z·∇u)dσ=z·νWeudσ2∂D∂D(5.2.14)!−(2W+z·∇W)euDwhereνistheoutwardnormalvectorto∂D.Proof.AsusualinderivingPohozaev-typeidentities,wemultiplyequation(5.2.1)byz·∇uanintegrateoverDtoobtain!!−(z·∇u) u=Weuz·∇u.(5.2.15)DDWeshallexpandeachsideof(5.2.15).Infact,bydirectinspection,itisnotdifficulttoverifytheidentity|∇u|2u(z·∇u)=div(∇u(z·∇u))−divz,2 1805TheAnalysisofLiouville-TypeEquationsWithSingularSourcesand(viatheGreen–Gausstheorem)obtaintheleft-handsideof(5.2.14).Concerningtheright-handsideof(5.2.14),wefind!!!!!Weuz·∇u=Wz·∇eu=div(zWeu)−2Weu−(z·∇W)euD!DD!!DD=(z·ν)Weudσ−2Weu−(z·∇W)eu,∂DDDand(5.2.14)isestablished.InthespecialcasewhereWisgivenby(5.2.2),wecanfurtherexpand(5.2.14)andconclude:Corollary5.2.12Letu∈C2(B1)satisfy(5.2.1)inB1where(5.2.2)holdswithV∈W1,+∞(B1).Then,foreveryr∈(0,1),wehave!!1222α+1ur|∇u|−(ν·∇u)dσ−rVedσ2{|z|=r}!!{|z|=r}(5.2.16)=−2(α+1)|z|2αVeu−|z|2α(z·∇V)eu.{|z|≤r}{|z|≤r}5.3BasicanalyticalfactsHere,weaimtoderivesomepreliminaryfactsconcerningasequenceuksatisfying−uukk=Wkein,(5.3.1)whereWkisafamilyofweightfunctionsand⊂R2isaboundedopenregulardomain.Animportantstartingpointforourdiscussionisgivenbythefollowing:Proposition5.3.13Letuksatisfy(5.3.1)andassumethat+(i)#Wk#L∞!()+#uk#L1()≤C,forsuitableC>0,|Wuk(ii)limsupk|e<4π.k→+∞Thenu+isuniformlyboundedinL∞().klocThisresulthasbeenobtainedbyBrezis–Merlein[BM],andholdswithinamoregen-eralLp-framework,whereassumptions(i)and(ii)arereplacedby!+uk4π#Wk#Lp()+#uk#L1()≤Candlimsup|Wk|e<,k→+∞pwith10andk0∈N,wehaveWkeuk≤4π−ε0,∀k≥k0.Therefore,weDcanapplyLemma5.2.1andRemark5.2.2tou2,ktoconclude#e|u2,k|#p≤C,L(D)forsuitablep>1andC>0.Inparticular,fromtheestimateabove,itfollowsthatu2,kisuniformlyboundedin1(D).Sinceu+≤u++|u+1L2,k|,wealsogetauniformboundforuinL(D).The1,kk1,kmeanvaluetheoremthenimpliesthatu+isactuallyuniformlyboundedinL∞(D).1,klocMoreover,weconcludethatWukpkeisuniformlyboundedinLloc(D),forsuitablep>1.Consequently,u2,kisuniformlyboundedinL∞(D)andthedesiredconclu-locsionfollows.Following[BM]wegivethefollowing:Definition5.3.14Apointz0∈iscalledablow-uppointforthesequenceukin,ifthereexistsasequence{zk}⊂suchthatzk→z0,andlimuk(zk)=+∞.k→+∞Inthesequel,weshalldenotebySthesetofblow-uppoints,andrefertoitastheblow-upset.AsaconsequenceofProposition5.3.13,wefind:Corollary5.3.15Supposethatuksatisfies(5.3.1)withWksuchthat!1#Wk#L∞()+≤C,forsomeq>0.(5.3.2)|Wk|q(i)If!|Wuk+∞limsupk|e<4π,thenuisuniformlyboundedinL().(5.3.3)klock→+∞ 1825TheAnalysisofLiouville-TypeEquationsWithSingularSources(ii)Ifz0∈isablow-uppointforuk,then!|Wukliminfk|e≥4π,(5.3.4)k→+∞Bδ(z0)foreveryδ>0sufficientlysmall.Furthermore,if,forasuitableconstantC>0,wehave!|Wukk|e0.Inthisway,wecancheckthevalidityoftheassumption(i)ofProposition5.3.13inBδ(z0),andconcludethatifz0∈isablow-uppointforuk,(henceitisablow-uppointforanyofitssubsequences),thennecessarily!|Wukliminfk(x)|e≥4π.k→+∞Bδ(z0)Therefore,when(5.3.5)holds,onlyafinitenumberofsuchblow-uppointsareallowed.Remark5.3.16Tojustifythenatureofassumptions(5.3.2),notethatwhen00sufficientlysmall,sothatB¯δ(z0)∩S1isemptyandz0istheonlypointofinBδ(z0).Notethatmaxu+→+∞,ask→+∞.kBδ(z0)+Infact,ifonthecontrary,(alongasubsequence)wesuppose#uk#L∞(Bδ(z0))≤C,thenforeveryε∈(0,δ),wehave!|Wuk2k|e=O(ε)→0,asε→0,Bε(z0)incontradictiontothefactthatz0∈.Thus,takingasubsequence,wecanletzk∈Bδ(z0):uk(zk)=maxuk→+∞,Bδ(z0)andzk→z1∈Bδ(z0).+isuniformlyboundedinL∞(),thenzSince,u1=z0necessarily,andklocsoz0isablow-uppointforsuchanewsubsequence,whoseblow-upsetS2containsthesetS1∪{z0}.Wearefinishedwhen=S2.Otherwise,weiteratetheargumentabovetoobtainanewsubsequencewhoseblow-upsetcontainsanadditionalpointof.Sincethenumberofelementsofisfinite,thisproceduremuststopafterafinitenumberofsteps,forwhichwearriveatthedesiredsubsequence{un},havingaskblow-upset.OurnexttaskistoinvestigatethenatureofthelimitingmeasureνofthesequenceunWnekinrelationtotheblow-upsetS.Inthisdirection,arathercompleteanalysisiskavailableforthenon-vanishingcase(namely,when(5.3.7)and(5.3.8)hold)throughtheworkofBrezis–Merle[BM],Li–Shafrir[LS],Brezis–Li–Shafrir[BLS],Li[L2],andChen–Lin[ChL1]and[ChL2].Butforourapplicationstothevortexproblem,weneedtoconsiderthesituationwhereWkactuallyvanishesatablow-uppoint,inotherwords,whenablow-uppointandavortexpointcoincide,withsaytheorigin.Withthisinmind,forgivenαk>0,wetakethefunctionWkoftheform,Wk(z)=|z|2αkVk(z),anddevotethenextsectiontothecorresponding“local”analysisaroundtheorigin. 1845TheAnalysisofLiouville-TypeEquationsWithSingularSources5.4Aconcentration-compactnessprincipleInthissectionwewishtoinvestigatethebehaviorofasequenceuksatisfying−u2αkukk=|z|VkeinB1(0),(5.4.1)forthecasewhenitadmitsablow-uppointattheorigin.Weassumethat00),weknowalreadybytheresultsoftheprevioussectionthatthefollowingholds:Corollary5.4.18!Letuksatisfy(5.4.1)andsupposethat(5.4.2)and(5.4.3)hold.|z|2αkVuk+∞(i)Iflimsupk(z)e<4π,thenuisuniformlyboundedinL(B1(0)).klock→+∞B1(ii)Ifzeroisablow-uppointforuk,then!liminf|z|2αkVukk(z)e≥4π,(5.4.4)k→+∞Bδforanyδ∈(0,1].Inaddition,if!|z|2αkVukke≤C,forsomeδ0∈(0,1],(5.4.5)Bδ(0)0withasuitableconstantC>0,thenzeroistheonlyblow-uppointforukinBδ(0),forsuitable,sufficientlysmallδ∈(0,δ0).Observethatproblem(5.4.1)enjoysanicescale-invariancepropertyasfollows:ifuksatisfies(5.4.1),thenforanyλ>0,thefunction
z1uk,λ(z)=uk+2(α+1)logλλ(5.4.6)satisfiesproblem(5.4.1)in
!Bλ={z:|z|<λ!},withVkreplacedbyVk,λ(z):=Vkzand|z|2αkVkeuk=|z|2αkVk,λeuk,λ.λB1BλBymeansofsuchinvariance,wewillbeabletouseablow-uptechniquetoimproveandcompletetheresultaboveinvariousdirections.5.4.1Theblow-uptechniqueWestartbyshowinghowtoimprove(5.4.4)undertheadditionalcondition:V0k→VuniformlyinCloc.(5.4.7) 5.4Aconcentration-compactnessprinciple185Proposition5.4.19Letuksatisfy(5.4.1)andassumethat(5.4.2),(5.4.3),and(5.4.7)hold.Ifzeroisablow-uppointforuk,then!liminf|z|2αkVukk(z)e≥8π,∀δ∈(0,1].(5.4.8)k→+∞Bδ(0)Proposition5.4.19shouldbecomparedwithanequivalentresultestablishedbyLi–Shafririn[LS]forsolution-sequences{uk}of(5.3.1)andsatisfying:!Wukk→WuniformlyinCloc()ande≤C.(5.4.9)Proposition5.4.20([LS]).Letuksatisfy(5.3.1)andassume(5.4.9).Ifz0∈isablow-uppointforuk,then!liminfWukke≥8π,∀δ>0small.(5.4.10)k→+∞Bδ(z0)Notethatthesecondconditionin(5.4.9)permitsustoverifyassumption(i)ofPropo-sition5.3.13;andthisimpliesthat(5.3.4)holdsforuk.Hence,Propositions5.4.19and5.4.20aimtoimprovethelowerboundin(5.4.4)and(5.3.4),respectively,whenthesequenceofweightfunctionsadmitsauniformlimit.Clearly,Proposition5.4.20coversasaparticularcaseProposition5.4.19,whenthelimitingfunctionWin(5.4.9)satisfies00.Assumethatξksatisfies⎧⎪⎪− ξk=Uk(z)eξkinDk={|z|0anda≥0.k→µ|z|locThenthefollowingholds:(a)ξkisuniformlyboundedinL∞();loc(b)alongasubsequence,2,β2ξk→ξuniformlyinC(R),β∈(0,1)(5.4.11)locandξsatisfies⎧⎨− ξ=µ|z|2aeξinR2,!2aξ(5.4.12)⎩|z|e<+∞,R2withξ(z0)=0. 1865TheAnalysisofLiouville-TypeEquationsWithSingularSourcesRemark5.4.221)Obviouslytheconditionξk(yk)=0isa“normalization”condi-tion,andinfact,theconclusionabovefollowsunderthemoregeneralrequirement:limsup|ξk(yk)|<+∞.k→+∞2)WecanapplytheclassificationresultinCorollary2.2.2,i.e.,(2.2.17)withα=a,toseethatξin(5.4.12)takestheform:λ01ξ(z)=log,γa=(5.4.13)(1+λ0µγa|za+1−y0|2)28(1+a)2with!µ|z|2aeξ=8π(1+a);(5.4.14)R2inaddition,ifa∈(0,+∞)N,theny0=0necessarily.(5.4.15)Moreover,in(5.4.13)theparametersy0∈Candλ0∈R+areconstrainedbytheconditionξ(z0)=0.Thisimplies:11
,sa+10:=γaµ|z−y0|≤andλ0=1−2s0±1−4s0≥1.(5.4.16)042s20Attimes,itisknownthatξ(z0)=maxξ=0.Forsuchcases,(5.4.13)holdswithR2ya+10=zandλ=1.0ProofofLemma5.4.21.Letfk=Ukeξk,sothataccordingtoourassumptions,fkisuniformlyboundedinL∞(R2).Furthermore,foreveryR>0,wecanuseCorol-loclary5.2.4,toobtainβ∈(0,1),independentofR,suchthatsupξk≤βinfξk+CR,(5.4.17)BRBRforsuitableCR>0.SinceforRsufficientlylarge,supξk≥ξk(yk)=0,wecanuse(5.4.17)toensureBRthatξkisalsoboundedfrombelowinBR,anduniformlysoink.Inotherwords,ξkisuniformlyboundedinL∞(R2).Wecanthenusestandardellipticregularitytheorytoloc2,γ2extendsuchuniformboundstoholdinC(R),forsomeγ∈(0,1).locHence,byadiagonalprocess,weobtainasubsequence(denotedinthesameway)suchthat2,α(R2),(5.4.18)ξk→ξ,uniformlyinClocwithξsatisfying(5.4.12)andξ(z0)=0asclaimed.Tosimplifynotationandwithoutlossofgenerality,fromnowonwesupposethat(5.4.7)issatisfiedwithV(0)=1.(5.4.19) 5.4Aconcentration-compactnessprinciple187ProofofProposition5.4.19.Toestablish(5.4.8),weonlyneedtoconsiderthecasewhere(5.4.5)holds.Thus,forδ>0sufficientlysmall,wecanassumethatzeroistheonlyblow-uppointforukinB¯δ.Lettingzk∈B¯δ,uk(zk)=maxuk,(5.4.20)B¯δweifnecessary,takeasubsequencesothat,uk(zk)→+∞andzk→0.(5.4.21)Setuk(zk)−εk=e2(αk+1)→0.Wenowdistinguishtwocases:Case1:zk=O(1),ask→+∞.(5.4.22)εkInthiscase,letusassumethatzkyk:=→z0,(5.4.23)εkholdsalongasubsequence.Setξk(z)=uk(εkz)+2(αk+1)logεk.(5.4.24)Notethat,ξk(yk)=maxξk=0.Thus,weeasilycheckthatξksatisfiesallas-{|z|≤δ}εkδ→+∞,andUsumptionsofLemma5.4.21,withykin(5.4.23),Rk=k(z)=εk|z|2αkVk(εkz)→|z|2αuniformlyinC0(R2),ask→∞.locConsequently,ξ∞(R2)(5.4.25)kisuniformlyboundedinLlocandalongasubsequence,ξ2(R2)(5.4.26)k→ξuniformlyinCloc'withξsatisfying(1.3.41)forµ=1anda=α.Inparticular,ξsatisfies:2|z|2αeξ=R8π(1+α)(see(5.4.14)).Therefore,byFatou’slemma,wefind:!!!|z|2αkVukξk2αξlimk(z)e=limUke≥|z|e=8π(1+α),k→+∞k→+∞{|z|≤δ}{|z|≤δ/εk}R2andthedesiredconclusionfollowsinthiscase. 1885TheAnalysisofLiouville-TypeEquationsWithSingularSourcesRemark5.4.23Noticethatinviewof(5.4.25),when(5.4.22)holds,thenmaxuk−B¯δuk(0)=−ξk(0)=O(1).Thatis,uk(0)=maxuk+O(1),inthiscase.B¯δCase2:|zk|→+∞,ask→+∞.εkuk(zk)
α−kInthissituation,setτe2εk→0,ask→+∞.Letk=|zk|αk=εk|zk|ξk(z)=uk(zk+τkz)−uk(zk)andzkτk2αkUk(z)=+zVk(zk+τkz)|zk||zk|inDδk={|z|≤}.2τkThen,⎧⎪⎪− ξk=Uk(z)eξkinDk,⎪⎨ξk(0)=0=maxξk,'Dk⎪⎪⎪⎩Uk(z)eξk≤C,Dkτk→0,weseethatU02forlargek.Sincek(z)→1uniformlyinC(R).Therefore,|zk|locwecanapplyLemma5.4.21toξkwitha=0,andbyRemark5.4.22,concludethatalongasubsequence,!122ξξk→ξ(z)=log
2uniformlyinCloc(R),ande=8π.1+1|z|2R28Consequently,!!!lim|z|2αkVkeuk≥lim|z|2αkVkeuk≥limUkeξkk→+∞k→+∞k→+∞Bδ(0){|z−zk|≤δ/2}Dk!≥eξ=8π.Therefore(5.4.8)holdsinthiscaseaswell.Forcompleteness,wealsoinclude:ProofofProposition5.4.20.Weproceedsimilarly,andletδ>0sufficientlysmallsuchthatz0istheonlyblow-uppointforukinB2δ(z0).Byatranslation,wecanalwaysassumethatz0=0.Letzk∈B2δ(0)suchthatuk(zk)=maxuk.Wehave{|z|<2δ}zk→0anduk(zk)→+∞.Settinguk(zk)−εk=e2→0,ask→+∞ 5.4Aconcentration-compactnessprinciple189and()δξk(z)=uk(zk+εkz)+2logεk,z∈Dk=|z|<εkweseethat⎧− ξξk⎨k=UkeinDk,!ξξk⎩k(0)=maxξk=0ande≤C,DkDkwithUk(z)=Wk(zk+εkz)→W(0)uniformlyinC0.Exactlyasabove,suchlocconditionssufficetofindasubsequence(denotedinthesameway),suchthatξk→ξ2,αuniformlyinCandξsatisfies:loc⎧⎪⎪− ξ=W(0)eξ,⎪⎨ξ(0)=maxξ=0,!R2⎪⎪⎪⎩ξe<+∞.R2Since ξ(0)≤0,weseethatnecessarilyW(0)≥0.Ontheotherhand,W(0)=0wouldimplyξ=constant,incontradictionwiththeintegratabilityofeξinR2.Therefore,1µ=W(0)>0andξ(z)=logµ2.1+|z|28Inparticular,!!!Wukξkξliminfke=liminfUke≥W(0)e=8π,k→+∞Bδ(0)k→+∞DkR2and(5.4.9)isestablished.TheproofofProposition5.4.19leavesustowonderwhenCase1actuallyoccurs.Moregenerally,whencanwereplace(5.4.8)withtheimprovedlowerbound!|z|2αkVukliminfke≥8(1+α),(5.4.27)k→+∞Bδ(0)foreveryδ∈(0,1]?Inthisrespect,afirstsimpleobservationpointstowardsaconditionthatplaysarelevantroleinthesequel.Corollary5.4.24InadditiontotheassumptionsofProposition5.4.19,supposethatsup{uk(z)+2(αk+1)log|z|}≤C,(5.4.28)|z|≤δ0forsuitableδ0∈(0,1)andC>0.Then(5.4.27)holds.Furthermore,if(5.4.5)isalsosatisfied,thenuk(0)=maxuk+O(1)(5.4.29)|z|≤δ0 1905TheAnalysisofLiouville-TypeEquationsWithSingularSourcesuk(zk)−Proof.Letzk∈B¯δ:uk(zk)=maxuk→+∞andεk=e2(αk+1).0|z|≤δ0Itsufficestoobservethatinviewof(5.4.28),zk→0necessarilyand(5.4.22)holds.Thisimmediatelyyieldsto(5.4.27).Moreover,ifwerecallRemark5.4.23,thenwealsoconclude(5.4.29).When(5.4.28)failstoholdforeveryδ0∈(0,1),wecanrefineCorollary5.4.24andProposition5.4.19asfollows:Lemma5.4.25Letuksatisfy(5.4.1)andassumethat(5.4.2)and(5.4.3)hold.Supposethereexistsasequence{zk}⊂B1{0}suchthatzk→0,uk(zk)+2(αk+1)log|zk|→+∞,then!liminf|z|2αkVukke≥4π,(5.4.30)k→+∞Bδ|zk|(zk)∀δ>0.Ifinadditionweassume(5.4.7),then!|z|2αkVukliminfke≥8π,(5.4.31)k→+∞Bδ|zk|(zk)∀δ>0.Proof.Itsufficestoprove(5.4.30)and(5.4.31)forδ∈(0,1).Tothispurpose,letkbesufficientlylarge,suchthatu1,k(z)=uk(zk+|zk|z)+2(αk+1)log(|zk|)iswell-definedinB1andsatisfies−u1,k(z)=|z|2αkV1,k(z)eu1,kinB1,2αkzk+zVwithV1,k(z)=k(zk+|zk|z).Noticethatzerodefinesablow-uppoint|zk|foru1,ksincewehaveu1,k(0)=uk(zk)+2(αk+1)log|zk|→+∞,ask→+∞.SinceV1,kisuniformlyboundedfromaboveandfrombelowawayfromzeroinBδ(0),∀!δ∈(0,1),wecanapplyCorollary5.4.18(ii)tou1,kandconcludethatliminfV1,keu1,k≥4π,∀δ∈(0,1).k→+∞Bδ(0)Asimplechangeofvariablesyieldsto(5.4.30).Ontheotherhand,when(5.4.7)holds,wecanapplyProposition5.4.19toanysubsequenceofu1,kandsimilarlyderive(5.4.31). 5.4Aconcentration-compactnessprinciple191Atthispoint,itappearsclearthatinordertoanalyzethebehaviorofukaroundtheblow-uppointzero,wemustanalyzeseparatelythecasewhere(5.4.28)holds,andthecasewhereitfails.Forthispurpose,itisusefultohaveavailablethefollowingalternative:Proposition5.4.26Letuksatisfy(5.4.1),andassumethat(5.4.2),(5.4.3),and(5.4.5)
hold.Thereexistconstantsε10∈0,andC>0,suchthat,alongasubsequence2thefollowingalternativeholds:(i)eithersup{uk(z)+2(αk+1)log|z|}≤C;(5.4.32)0<|z|≤2ε0(ii)orthereexistsasequence{zk}⊂B1{0}suchthatzk→0,uk(zk)+2(αk+1)log|zk|→+∞,(5.4.33)sup{uk+2(αk+1)log|z|}≤C.(5.4.34)0<|z|≤2ε0|zk|Proof.Takingintoaccount(5.4.5),set!β=limsup|z|2αkVukke.(5.4.35)k→+∞Bδ0
0,1,thenwefindasequencezIf(5.4.32)failstoholdforeveryε0∈1,k→0anda2subsequenceofuk,whichwedenoteinthesameway,suchthatuk(z1,k)+2(αk+1)log|z1,k|→+∞.Thus,byLemma5.4.25weknowthat:!|z|2αkVukliminfke≥4π,k→+∞Bδ|z1,k|(z1,k)∀δ>0.Repeatthealternativeaboveforthesequence:u1,k(z)=uk(|z1,k|z)+2(αk+1)log|z1,k|.
Therefore,either(i)holdsforsuitableε0,1,andleadsto(5.4.33),(5.4.34)0∈2withzk=z1,k;orthereexistsasecondsequencez2,k∈B1{0},suchthat,alongasubsequence,thereholdsz2,k→0,uk(z2,k)+2(αk+1)log|z2,k|→+∞,z1,k!liminf|z|2αkVukke≥4π,∀δ>0.k→+∞Bδ|z2,k|(z2,k) 1925TheAnalysisofLiouville-TypeEquationsWithSingularSourcesNotethatthesetsBδ|z1,k|(z1,k)andBδ|z2,k|(z2,k)donotintersecteachotherforδ∈(0,1)andklarge.Therefore,inthiscaseweseethatβin(5.4.35)mustsatisfy:β≥8π.Wemayalsorepeatthealternativeabovefortheseconditeratedsequenceu2,k(z)=uk(|z2,k|z)+2(αk+1)log|z2,k|,andsoon.Observethateachtimesuchaniteratedsequencefails
toverify(5.4.32),∀ε0,1,wecontributewithanamountof(atleast)4πtothe0∈2valueβin(5.4.35).Thus,afterafinitenumberofsteps,wemustendupwithaniteratedsequencethatsatisfies(5.4.32)forsomeε0∈(0,1/2).Thisfact,whenexpressedfortheoriginalsequenceuk,yieldstothedesiredpropertiesof(5.4.33)and(5.4.34).5.4.2AConcentration-Compactnessresultarounda“singular”pointAmoreelaborateanswertothequestionconcerningthevalidityof(5.4.27)requirestheintroductionofthefollowingsuitableboundaryconditionsonuk:supuk−infuk≤C,δ0∈(0,1],(5.4.36)∂Bδ∂Bδ00forsuitableC>0.Asweshallsee,thebehaviorofukaroundtheorigin(ablow-uppoint)isseriouslyaffectedbythevalidityof(5.4.36).Tothisend,wealsoneedtostrengthen(5.4.2)byrequiringthatVkisdifferentiableinB1(0)where|∇Vk|≤A.(5.4.37)Proposition5.4.27InadditiontotheassumptionsofProposition5.4.19,supposefur-therthat(5.4.37)holdsandforsomeδ0∈(0,1]andproperty(5.4.36)issatisfied.Then(5.4.27)holds.TheproofofProposition5.4.27,aswellasotherinterestingconcentration/compact-nessproperties,areaconsequenceofthefollowingresultestablishedin[BT2].Theorem5.4.28Letuksatisfy(5.4.1),andassume(5.4.2),(5.4.3),(5.4.5),(5.4.36),and(5.4.37).Ifzeroisablow-uppointforuk,thenthereexistsr0∈(0,1]suchthatalongasubsequence2αkVuk|z|ke8π(1+α)δz=0,weaklyinthesenseofmeasureinBr.0BeforegivingtheproofofTheorem5.4.28,weshallderivesomeofitsinterestingconsequences.ProofofProposition5.4.27.Again,weonlyhavetoconsiderthecasewhere:!|z|2αkVukke≤C.{|z|≤δ0}Therefore,weseethatuksatisfiesallassumptionsofTheorem5.4.28,andsowefindr0∈(0,1),suchthatalongasubsequence 5.4Aconcentration-compactnessprinciple193!|z|2αkVukk(z)e→8π(1+α),Br∀r∈(0,r0),and(5.4.27)follows.Proposition5.4.29Letuksatisfy(5.4.1).Assume(5.4.2),(5.4.3)holdandforα>0in(5.4.3),alsoassume(5.4.37)holds.Ifuk≥−MinB1(0),thenukisuniformlyboundedinL∞(B1(0)).locProof.Byreplacingukwithuk+M,wecanalwaysassumethatuk≥0inB1(0).Claim:Forevery⊂⊂B1(0),thereexistsaconstantC(dependingon)suchthat!|z|2αkVukke≤C.(5.4.38)Toestablish(5.4.38),wefollowanargumentgivenbyBrezis–Merlein[BM].Let´ϕ1bethefirstpositiveeigenfunctionof−inH1(B1(0)),anddenotebyλ1thecor-0'respondingeigenvalue.Wenormalizeϕ1tohaveϕ1=1.WemultiplyequationB1(5.4.1)byϕ1andintegrateoverB1toobtain:!!!!|z|2αkVuk∂ϕ1keϕ1=λ1ukϕ1+uk≤λ1ukϕ1.(5.4.39)B1∂νB1∂B1B1Ontheotherhand,by(5.4.2)andwiththehelpofJensen’sinequality(2.5.8),wehave!!''|z|2αkVkeukϕ1≥b1euk+2αklog|z|ϕ1≥b1eB1ukϕ1+2αkB1log|z|ϕ1.B1B1Thus,from(5.4.39)wederive''!!uϕ12αlog1ϕeB1k1≤λ1ekB1|z|1ukϕ1≤Cukϕ1,b1B1B1whichimplies!ukϕ1≤2C.(5.4.40)B1(0)Inserting(5.4.40)into(5.4.39),wearriveat(5.4.38).Now,letusarguebycontradictionandassumethatukadmitsablow-uppointz0inB1.Asaconsequenceof(5.4.38),wefindδ0>0sufficientlysmall,sothatz0istheonlyblow-uppointforukinBδ(z0)⊂B1.Soukisuniformlyboundedin02,γC(Bδ(z0){z0}),andwecanpasstoasubsequencetoderive:loc02αkVuk|z|keν,weaklyinthesenseofmeasureinBδ(z0);02(Bukξ,uniformlyinCδ(z0){z0});loc0and− ξ=νinthesenseofdistributionsinBδ(z0).0 1945TheAnalysisofLiouville-TypeEquationsWithSingularSourcesInviewofCorollary5.4.18(ii),ν(z0)≥4πnecessarily.Thisleadstotheestimate:1ξ(z)≥2log−C,inBδ(z0).(5.4.41)0|z−z0|Ifz0=0,itsufficestohaveacontradiction.Infact,inthiscase,weknowfrom(5.4.38)andFatou’slemmathateξisintegrableinBδ(z0);butisimpossiblebyvirtueofestimate(5.4.41).Notethatwhenα=0in(5.4.3),theargumentaboveleadstoacontradictionalsoincasez0=0.Hence,supposeα>0andz0=0.Inthissituation,Fatou’slemmaimplies!|z|2αeξ<+∞,(5.4.42)Bδ(0)∀δ∈(0,δ0).Ontheotherhand,sinceuk≥0andzeroistheonlyblow-uppointforukinB¯δ,wemaycheckthat(5.4.36)holds.SoweareinapositiontoapplyTheorem05.4.28andconcludethatν(0)≥8π(1+α).Consequently,1ξ(z)≥4(α+1)log−CinBδ,0|z|whichclearlycontradicts(5.4.42).Remark5.4.30Bydirectinspectionoftheproofabove,weseethatwhenα∈(0,1],condition(5.4.37)canbeweakenedto(5.4.7)(seeProposition5.4.19).Corollary5.4.31UndertheassumptionsofProposition5.4.29,ifukblowsupinB1(0)theninfuk→−∞,ask→+∞.B1Againobservethat,ifweknowapriorithatzeroisnotablow-uppoint,thentheconclusionofCorollary5.4.31followswithoutrequiring(5.4.37).Proposition5.4.32Letuksatisfy(5.4.1)andsupposethat(5.4.2),(5.4.3),and(5.4.5)hold.Inaddition,assume(5.4.37)ifandonlyifα>0in(5.4.3).Thereexistsr0∈(0,1]andasubsequenceofuk(denotedinthesameway),forwhichonlyoneofthefollowingalternativeshold:(a)ukisboundeduniformlyinL∞(Br);loc0(b)supuk→−∞,forevery⊂⊂Br;0(c)thereexistszk→0,withuk(zk)→+∞,suchthatui.supk→−∞,∀⊂⊂Br{0},0ii.|z|2αkVkeukβδz=0weaklyinthesenseofmeasuresinBr,andβ≥4π.0Proof.Withoutlossofgeneralitywecanassumethat(5.4.5)holdsforδ0=1.Indeed,ifthiswasnotthecase,thenbymeansof(5.4.6)wewouldsimplyreplaceukwith 5.4Aconcentration-compactnessprinciple195uk(δ0z)+2(αk+1)logδ0.WecanapplyProposition5.3.17toukinB1,andfindasubsequenceofuk,whichforsimplicitywedenoteinthesameway,suchthat+isuniformlyboundedinL∞(Bu1S),(5.4.43)klocwhereSistheblow-upset(possiblyempty)of(thesubsequence)uk.Wecanalsoassumethat2αkVuk|z|keν,weaklyinthesenseofmeasuresinB1,(5.4.44)withνafinitemeasureinB1.Notethat,inviewof(5.4.43),anyothersubsequenceofukadmitsthesameblow-upsetS.Forthecasewhentheblow-upsetSisempty,ukisuniformlyboundedfromaboveinanysubsetofB1.ThenwecanuseProposition5.4.29togetherwithCorollary5.2.9toconcludethat,alongapossiblesubsequence,eitheralternative(a)or(b)holds.SupposenowthatSisnotempty,butcontainsafinitenumberofpoints.Thenforanyδ>0sufficientlysmallandz0∈S,wecanapplyCorollary5.4.31tou˜k(z)=uk(z0+δz)+2(αk+1)logδ,z∈B1,andseethatinfuk=infuk→−∞,ask→+∞,(5.4.45)∂Bδ(z0)Bδ(z0)wherein(5.4.45)wehaveusedthesuperharmonicityofuk.Consequently,foreveryδ>0sufficientlysmall,thereholdsinfuk→−∞,ask→+∞,(5.4.46)1,δwhere21,δ=B1Bδ(p)⊂⊂B1S.p∈SThus,bymeansof(5.4.43),wecanapplyCorollary5.2.9andconclude:supuk→−∞,ask→+∞.(5.4.47)1,δAtthispointtheremainingpartoftheprooffollowseasily.Indeed,wecanfindr0>0suchthateitherBr(0)∩SisemptyorBr(0)∩S={0}.Inthefirstcase,by(5.4.47)00weseethatalternative(b)holds.Whileinthesecondcase,weeasilycheckthevalidityofparti.ofproperty(c),andseethatthemeasureνin(5.4.44)issupportedexactlyattheorigininBr.Namely,ν=βδz=0inBrandβ≥4πbyvirtueof(5.4.4).00Remark5.4.33Ifweassume(5.4.7),thenwecanusePropositions5.4.19and5.4.20todeducethatpartii.ofproperty(c)inProposition5.4.32holdswithβ≥8π. 1965TheAnalysisofLiouville-TypeEquationsWithSingularSourcesWeshallpresentexamplesinSection5.5.5belowshowingthattheconditionβ≥8πissharpandcannotbeimprovedingeneral.Thisissurprisinginaway,expeciallyifwetakeintoaccountTheorem5.4.28,whereclearlytheadditionalcondition(5.4.36)mustplayacrucialrole.Theroleof(5.4.36)towardsthe“concentration”phenomenonwaspointedoutfirstbyWolanskyforthenon-vanishingcase,i.e.,whenαk=0in(5.4.1)(seealso[L2]).Thegeneralcasewasderivedin[BT2]bymeansofthePohozaev’stypeidentity,givenin(5.2.16).Thisapproachappearsparticularlyusefulwhencondition(5.4.36)holdsanditwillbeverymuchexploitedbelow.Seealso[OS1]and[OS2]forrelatedresults.ProofofTheorem5.4.28.Bytakingasubsequenceifnecessary,wecanassumethat(5.4.44)holds,withνafinitemeasureinB1(0),satisfying:ν(0)=β≥8π.Furthermore,thereexistsr0∈(0,δ0],suchthatzeroistheonlypointofblow-upforukinBr.Asaconsequence,wefindthatfk=|z|2αkVkeukisuniformlybounded0inC0(Br{0}).Therefore,inviewofassumption(5.4.36),wecanuseGreen’sloc0representationformulaforϕk=uk−infuk,∂Bδ0toobtainthatβ12ϕk→ϕ=log+φ,uniformlyinC(Br{0}),(5.4.48)2π|z|loc0withφaregularfunctioninBr.0Noteinparticularthatβz∇uk=∇ϕk→∇ϕ=+∇φ,(5.4.49)2π|z|2uniformlyinC1(Br{0}).loc0Fixr∈(0,r0),andusePohozaev’sidentity(5.2.16)forukinBrtoobtain:!|∇uk|2−(ν·∇u2rk)dσ2∂Br!!!(5.4.50)=r|z|2αkVk(z)eukdσ−2(αk+1)|z|2αkVkeuk−(z·∇Vk)|z|2αkeuk.∂BrBrBrInviewofourassumptionsonVk,weeasilycheckthat!(z·∇V2αkukk)|z|e≤Cr,BrwithasuitableconstantC>0independentofk∈Nandofr. 5.4Aconcentration-compactnessprinciple197Ifwepasstothelimitin(5.4.50)andask→+∞,use(5.4.49),wefind!2limr|z|2αkVukβke=−+2(α+1)β+o(1),(5.4.51)k→+∞4π∂Brwitho(1)→0,asr→0.Claim:infuk→−∞,ask→+∞.(5.4.52)∂Bδ0Toestablish(5.4.52)wearguebycontradictionandsupposethatinfuk>−M,for∂Bδ0M>0asuitableconstant.Thus,bymeansofFatou’slemmaand(5.4.48),wefind:!!!C>limsup|z|2αkeuk≥e−Mlimsup|z|2αkeϕk≥e−M|z|2α−β/2πeφ.k→+∞k→+∞BrBrBr000Thisimpliesβ<4π(1+α).(5.4.53)Consequently,!!r|z|2αkVuk−M2αϕlimsupk(z)edσ≤er|z|Vedσk→+∞(5.4.54)∂Br∂Br≤Cr2(α+1)−β/2π→0,asr→0.Using(5.4.54)togetherwith(5.4.51),asr→0,wederiveβ2−+2(α+1)β=0,i.e.,β=8π(1+α),(5.4.55)4πincontradictionwith(5.4.53).Once(5.4.52)isestablished,wecanuse(5.4.48)toconcludethat,foreverycom-pactsetK⊂Br{0},0supuk→−∞.KThatis,|z|2αkVuk0ke→0,uniformlyinC(Br{0})(5.4.56)loc0andν=βδz=0.(5.4.57)Furthermore,(5.4.56)impliesthattheleft-handsideof(5.4.51)mustvanish.Soasr→0,wededucethatβ=8π(1+α)andwearriveatthedesiredconclusion. 1985TheAnalysisofLiouville-TypeEquationsWithSingularSources5.4.3Aglobalconcentration-compactnessresultInconcludingourdiscussionontheconcentration-compactenssphenomenon,weshowhowtopatchtogetherthe“local”informationderivedabovetoobtaina“global”concentration-compactnessresultasfollows.Motivatedbyproblems(5.1.1)and(5.1.2),weconsider⊂R2boundedopensetandwelet$m|z−z2αi,kWk(z)=i|Vk(z),z∈,(5.4.58)i=1with{z1,...,zm}⊂givendistinctpoints;αi,k→αi≥0,i=1,...,m;(5.4.59)00asuitableconstant.Wehave:Theorem5.4.34Letuksatisfy(5.4.61)andassume(5.4.58),(5.4.59),and(5.4.60)hold.Inaddition,if(5.4.58)holdswithαj>0forsomej∈{1,...,m},supposethat:Vk∈C0,1(Br(zj))and|∇Vk|≤AinBr(zj),forsuitabler0>0sufficientlysmall00andA>0.Alongasubsequence,oneofthefollowingalternativeholds:(a)ukisuniformlyboundedinL∞();loc(b)supuk→−∞,ask→∞,forevery⊂⊂;(c)thereexistsafinitesetS={q1,...,qs}⊂(ablow-upset)andcorrespondingsequences{zj,k}⊂suchthatask→∞(i.)zj,k→qjanduk(zj,k)→+∞,∀j=1,...,s,(ii.)usupk→−∞,forevery⊂⊂S,(5.4.62)(iii.)sWkuk→βjδq,(5.4.63)jj=1weaklyinthesenseofmeasurein,withβj≥4π,foreveryj=1,...,s.Inaddition,ifVk→VinC0()then(5.4.63)holdswithβj≥8π,j=1,...,s.loc 5.5Aquantizationpropertyintheconcentrationphenomenon199Proof.Clearly,wecanapplyProposition5.3.17tofindasubsequenceofuk,denotedinthesameway,suchthatu+isuniformlyboundedinL∞(S),(5.4.64)klocwhereSisthe(possiblyempty)blow-upsetofthesubsequenceofuk.Wealsomaysupposethatask→∞,Wuk→ν,weaklyinthesenseofmeasurein,(5.4.65)kewhereνisafinitemeasurein.AsintheproofofProposition5.4.32,weseethateitheruksatisfiesalternative(a),or:supuk→−∞,ask→∞,forevery⊂⊂S.Inparticular,ifSisemptytheneitheralternative(a)or(b)holds.Ontheotherhand,whenSisnotempty,itcontainsafinitenumberofpoints,sayS={q1,...,qs},forwhich(c)(i)and(c)(ii)hold.Hence,themeasureνin(5.4.65)issupportedinS.Now,forδ>0sufficientlysmallandq∈S,wecanapplyProposition5.4.32tou˜k=uk(q+δz)+2(αk+1)logδ,z∈B1;andweconcludethatν|Bδ(q)=βδqwithβ≥4π.Thus,(c)(iii)isestablishedoncewealsotakeintoaccountRemark5.4.33.5.5Aquantizationpropertyintheconcentrationphenomenon5.5.1PreliminariesThegoalofthissectionistogiveaprecisecharacterizationoftheconcentrationvalueβwithinthatoccursinalternative(c)ofProposition5.4.32.Forthispurpose,wetakeuktosatisfy2αkVuk−uk=|z|keinB1,(5.5.1)2αkVuk|z|keβδz=0,(5.5.2)weaklyinthesenseofmeasureinB1.ItfollowsfromTheorem5.4.28that,whenuksatisfiesalso(5.4.36),thennecessarilyβ=8π(1+α),(5.5.3)providedthat(5.4.2),(5.4.3)and(5.4.37)alsohold.ExplicitexamplesdiscussedinSection5.5.5belowshowthatwhen(5.4.36)isnotsatisfied,then(5.5.3)failstoholdingeneral.Ontheotherhand,suchexamplesalsoindicatethatinanycaseβcannottakeanyarbitraryvaluelargerthanorequalto8π,butinfactmustberestrictedtosatisfyingasortof“quantization”propertyasfollows:β∈8πN∪8π(N+α).(5.5.4) 2005TheAnalysisofLiouville-TypeEquationsWithSingularSourcesItisoneofthemaingoalsofthissectiontoprove(5.5.4)andthuscompleteTheorem5.4.34asfollows:Theorem5.5.35Ifalternative(c)holdsinTheorem5.4.34,thenproperty(iii)isveri-fiedwith,βj∈8πN,forqj∈{/z1,...,zm}orβj∈{8π(N+αi)}∪8πNforqj=zi,(5.5.5)forsomei=1,...,mandj=1,...,s.Ifwetakeαi,k=0in(5.4.58),then(5.5.5)givesβj∈8πN,∀j=1,...,s.ThissituationwashandledfirstbyLi–Shafririn[LS],whilethegeneralcasewasestablishedbytheauthorin[T5].Theorem5.5.35easilyfollowsonce(5.5.4)isestablished.Infact,wecanlocalizeouranalysisaroundeachblow-uppointqj,andafterasuitabletranslation,wecanscaleoursequenceaccordingto(5.2.13)toobtainaproblemofthetype(5.5.1)and(5.5.2),forwhichwheneverqj∈{/z1,...,zm},wetakeαk=0∀k.Letusmentionthatintheprocessofestablishing(5.5.4),wealsoobtainanin-equalityofthetype“sup+inf”inthesamespiritof[BLS],[ChL4],and[Sh].ThiswillbediscussedinSection5.5.3.5.5.2AversionofHarnack’sinequalityThroughoutthissectionweassumeαk≥0andαk→α,(5.5.6)andV0,1k∈C(B1):00,andasequence{zk}⊂B1suchthat2(i)zk→0,uk(zk)+2(αk+1)log|zk|→+∞;(5.5.8)(ii)sup{uk(z)+2(αk+1)log|z|}≤C0;(5.5.9)|z|≤2ε0|zk|(iii)!|z|2αkVukke≤C0.(5.5.10)|z|≤(1+ε0)|zk| 5.5Aquantizationpropertyintheconcentrationphenomenon201Letvk(z)=uk(|zk|z)+2(αk+1)log|zk|,(5.5.11)thenalongasubsequence,thefollowingalternativeholds:(a)eithermaxvk→−∞andinfuk≤maxvk+2(αk+1)log|zk|+C,{|z|≤ε0}B1{|z|≤ε0}(b)orvk(0)→+∞andinfuk≤−uk(0)+C,B1forasuitableconstantCdependingonlyonb1,b2,andAin(5.5.7).Proof.Tosimplifyournotation,wetakeαk=α,∀k∈N.Furthermore,bypassingtoasubsequenceifnecessary,from(5.5.7),wecanfurtherensurethat0(BVk→V,uniformlyinCloc1),(5.5.12)whereagain,weloosenogeneralitybyassumingthatV(0)=1.(5.5.13)Observethatvksatisfies:⎧⎪⎪− vk=|z|2αVk(|zk|z)evkinD={|z|≤1+ε0},⎪⎨'|z|2αVk(|zk|z)evk≤C0,(5.5.14)D⎪⎪⎪⎩sup{vk(z)+2(α+1)log|z|}≤C0.|z|≤ε0ThusinviewofCorollary5.4.24,alongasubsequence,weseethateithervk(0)=maxvk+O(1)→∞,ask→+∞,(5.5.15)|z|≤ε0ormaxvk0dependingonlyonk.Thus,wecanchooseλsufficientlynegative(dependingonk)suchthat∀µ≤λ,µωk(2µ−t,θ)−ωk(t,θ)<0,fort∈,0andθ∈[0,2π);2∂µωk(t,θ)>0,fort0specifiedaccordingto(5.5.22).8(α+1)BymeansofProposition2.2.3,weseethatforfixedθ∈[0,2π),thefunction1ω(·,θ)issymmetricwithrespecttotheaxist=log√1,withτ=λ0γ22(α+1).

ταNamely,ωlog√1−t,θ=ωlog√1+t,θ,∀t∈R,∀θ∈[0,2π).Moreoverττω(·,θ)isincreasingfortlog√1andattainsitsstrictττmaximumvalueatt=log√1.Noticealsothatτω(t,θ)≤2(α+1)t+logλ0.(5.5.34)Inviewof(5.5.20)and(5.5.21),foreveryfixeds∈R,wehavesup|ωk(t+logεk,θ)−ω(t,θ)|→0,ask→+∞.(5.5.35){t≤s,θ∈[0,2π)}Thusforlargek,wededucesup|ωk(t+logεk,θ)−ω(t,θ)|<1,(5.5.36){t≤4+log√1,θ∈[0,2π)}τand11ωk4+log√+logεk,θ<ωklog√+logεk,θ,∀θ∈[0,2π).(5.5.37)ττAsaconsequenceof(5.5.37)wecheckthat(5.5.29)failstoholdwhenλ=logεk+log√1+2,t=logεk+log√1+4,andkislarge.Fromthisfact,weimmediatelyττdeducetheestimate(5.5.32).Furthermore,using(5.5.36)and(5.5.34),forklargewecanestimate:ωk(2λk,θ)≤ω(2λk−logεk,θ)+1≤2(α+1)(2λk−logεk)+O(1)≤2(α+1)logεk+O(1)=−uk(0)+O(1).Therefore,using(5.5.31)and(5.5.27),wefindinfuk=minuk≤minωk(0,θ)≤maxωk(2λk,θ),B1∂B1θ∈[0,2π)θ∈[0,2π)≤−uk(0)+O(1),ask→+∞,and(5.5.33)follows.ConclusionoftheproofofTheorem5.5.36.NotethatbyvirtueofFact1andLemma5.5.37,thestatementinalternative(b)isestablished.Concerningalternative(a)wehave: 2065TheAnalysisofLiouville-TypeEquationsWithSingularSourcesClaim4:If(5.5.17)holds,thenλk≤log|zk|+O(1),ask→+∞.(5.5.38)Toestablish(5.5.38),noticethatfrom(5.5.25)and(5.5.26)wefindasuitableσ>0,suchthatforksufficientlylarge,wehave1v(yk+δkz)≤vk(yk)−2σ,≤|z|≤3,(5.5.39)2withykdefinedin(5.5.24).Letρk∈(0,+∞)andθk∈[0,2π)bethepolarcoordi-natesforyk;thatis,iθk=yρkek,soρk→1,ask→+∞.Since



ω222klog(1+s)ρk|zk|,θk=vk(1+s)yk+2(α+1)logρk(1+s)A2−|zk|ρk(1+s),b1∀s>0,wecanuse(5.5.39)todeduceωk(log|zk|+logρk+2log(1+δk),θk)<ωk(log|zk|+logρk,θk)−σ,(5.5.40)providedthatkissufficientlylarge.Thisshowsthat,forksufficientlylarge,λ=log|zk|+logρk+log(1+δk)andt=log|zk|+logρk+2log(1+δk)resultsinthefailureofinequality(5.5.29)toholdforθ=θk.Wethusconclude(5.5.38).Atthispoint,wearereadytoderivepart(a)ofourstatement.Indeed,from(5.5.31)wehaveinfuk=infuk=minωk(0,θ)+A/b1≤maxωk(2λk,θ)+A/b1B1∂B1θ∈[0,2π)θ∈[0,2π]e2λk+iθ=maxvk+2(α+1)(2λk−log|zk|)+A/b1(5.5.41)θ∈[0,2π)|zk|≤maxvk+2(α+1)log|zk|+C,|z|≤R0|zk|forsuitableconstantsR0andC.ThiscompletestheproofofTheorem5.5.36.Remark5.5.38Notethatinequality(5.5.31)containsaslightlystrongerstatementforalternative(a)ofTheorem5.5.36.5.5.3Inf+SupestimatesInthissectionwediscussaninterestingconsequenceofTheorem5.5.36,concerningsuitable“inf+sup”estimatesthatarevalidforsolutionsoftheequation:2αV(z)euinB−u=|z|1,(5.5.42)withVsatisfying,00and0∈Kthen(5.5.44)nolongersufficestoprovidea“global”estimateofthe(5.5.46)type.Inthiscase(5.5.44)givesaweakerstatementthan(5.5.46);sinceitispossibletoconstructasequenceukofsolutionsof(1.3.25),withVk=1,thatadmitszeroasablow-uppointinB1,andatthesametime,uk(0)→−∞,ask→+∞.WerefertoSection5.5.5belowfordetails.Therefore,thevalidityof(5.5.46)forα>0and0∈K,remainsachallengingopenproblem.Herewewillbeabletoprove(5.5.46)onlyundersomeadditionalas-sumptions(seeCorollary5.5.42andTheorem5.6.59).Inordertoestablish(5.5.44),weshallneedsomepreliminaryinformation.Wearegoingtoarguebycontradictionandassumethatthereexistsasequenceuksuchthat−u2αukk=|z|VkeinB1,(5.5.47)withVksatisfying(5.5.7),anduk(0)+infuk→+∞.(5.5.48)B1 2085TheAnalysisofLiouville-TypeEquationsWithSingularSourcesWithoutlossofgenerality,andbypassingtoasubsequenceifnecessarywecanfurtherassumethat(5.5.12)and(5.5.13)hold.Notethatuk(0)−εk:=e2(α+1)→0,ask→+∞,(5.5.49)aseasilyfollowsfrom(5.5.48).Lemma5.5.41Foragivenk∈N,thereexistsrk∈(0,1]suchthat!|z|2αVukke≤8π(1+α),(5.5.50){|z|≤rk}andrk→+∞.(5.5.51)εkProof.WeadaptanargumentofShafrir[Sh],alsousedin[BLS].Fixk∈N.If'|z|2αVkeuk≤8π(1+α),thenwejusttakerk=1.HencesupposethatB1!|z|2αVukke>8π(1+α),(5.5.52)B1andforr∈(0,1)define!1G(r)=uk(0)+ukdσ+4(α+1)logr.2πr∂BrWhence⎛⎞!!1∂uk4(α+1)1⎜⎟G(r)=+=⎝uk+8π(1+α)⎠2πr∂rr2πr∂BrBr⎛⎞!=1⎜|z|2αVuk⎟⎝8π(1+α)−k(z)e⎠.2πrBrTherefore,inviewof(5.5.52),thereexistsauniquerk∈(0,1)suchthat!|z|2αVukk(z)e=8π(1+α),(5.5.53)BrkandG(rk)=max{G(r),r∈(0,1)}.(5.5.54) 5.5Aquantizationpropertyintheconcentrationphenomenon209Usingthesuperharmonicityofuk,andasaconsequenceof(5.5.48)and(5.5.54),wefind:!12(uk(0)+2(α+1)logrk)≥uk(0)+ukdσ+4(α+1)logrk2πrk∂Brk=G(rk)≥G(r)≥uk(0)+infuk=uk(0)+infuk∂BrBr≥uk(0)+infuk→+∞,ask→+∞.B1Thusuk(0)+2(α+1)logrk→+∞,ask→+∞,andwederiverk1(u(0)+2(α+1)logr)=e2(α+1)kk→+∞,ask→+∞,εkasclaimed.ProofofTheorem5.5.39.Undertheassumptionofcontradictionby(5.5.48),setu1,k(z)=uk(rkz)+2(α+1)logrk,(5.5.55)withrkasgiveninLemma5.5.41.Observethat⎧()⎪⎪⎪⎨−u1,k=|z|2αVk(rkz)eu1,kinBk=|z|≤1,rk'(5.5.56)⎪⎪|z|2αVk(rkz)eu1,k≤8π(1+α),⎪⎩Bku1,k(0)→+∞,ask→+∞.Hence,accordingtoProposition5.4.26,wehavethatthefollowingalternativeholdsaroundtheorigin:eithersup{u1,k(z)+2(α+1)log|z|}≤C,(5.5.57)|z|≤2ε0or∃z1,k→0,u1,k(z1,k)+2(α+1)log|z1,k|→+∞,sup{u1,k(z)+2(α+1)log|z|}≤C,(5.5.58)|z|≤2ε0|z1,k|forsuitableε10∈(0,)andC>0.2Incase(5.5.57)holds,weuseCorollary5.4.24towrite:u1,k(0)=maxu1,k+O(1),ask→+∞.|z|≤2ε0 2105TheAnalysisofLiouville-TypeEquationsWithSingularSourcesConsequently,εkξk(z)=uk(εkz)+2(α+1)logεk=u1,kz−u1,k(0)rkrkεsatisfiesallassumptionsofLemma5.4.21forRk=0→+∞,yk=0andUk(z)=εk|z|2αVk(εkz)→|z|2αinC0(R2),ask→+∞.Soweconcludethat(5.5.20)andloc(5.5.21)holdforξk,andwecanapplyLemma5.5.37tofinduk(0)+infuk≤C,B1incontradictionwith(5.5.48).Ontheotherhand,if(5.5.58)holds,thenweareinapositiontoapplyTheorem5.5.36toukwithzk=rkz1,k.Sincealternative(b)immedi-atelyleadstoacontradictionof(5.5.48),wesupposethatvk(z)=uk(|zk|z)+2(α+1)log|zk|satisfiesmaxvk→−∞,(5.5.59)|z|≤ε0andsoinfuk≤maxvk+2(α+1)log|zk|+C.(5.5.60)B1{|z|≤ε0}Conditions(5.5.59)and(5.5.60)stillpermitustocontradict(5.5.48)asfollows:uk(0)+infuk≤vk(0)+maxvk+C≤2maxvk+C→−∞,B1{|z|≤ε0}{|z|≤ε0}ask→+∞.Andweconcludethevalidityof(5.5.44)asdesired.ByTheorem5.5.39,wecancheckthevalidityof(5.5.46)asfollows:Corollary5.5.42Foragivenc0>0,letusatisfy(5.5.42)whereVsatisfies(5.5.43)andsup{u(z)+2(α+1)log|z|}≤c0.B1Foreveryr∈(0,1),thereexistsaconstantC=C(r,α,b1,b2,A)suchthatsupu+infu≤C.(5.5.61)BrB1Proof.AsbeforewearguebycontradictionandbyvirtueofCorollary5.5.40wesupposetheexistenceofuksatisfying(5.5.47)withVkasin(5.5.7)andsuchthatthefollowingconditionshold:max{uk+2(α+1)log|z|}≤c0,(5.5.62){|z|≤1}andforasuitablesequence{zk}⊂B1,zk→0anduk(zk)+infuk→+∞.(5.5.63)B1 5.5Aquantizationpropertyintheconcentrationphenomenon211uk(zk)−Clearlyuk(zk)→+∞,andsoεk=e2(α+1)→0,while(5.5.62)implieszk=O(1),ask→+∞.(5.5.64)εkExactlyasinLemma5.5.41,property(5.5.63)allowsonetofindrk∈(0,1]suchthat!2αukrk|z|Vke≤8π(1+α)and→+∞,ask→+∞.εkBrk(zk)Using(5.5.64)weseethatzk→0,ask→+∞,(5.5.65)rk'andconsequently,|z|2αVkeuk≤8π(1+α),providedthatkissufficientlylarge.Brk(0)2Sinceby(5.5.7)wecanalwaysverify(5.5.12)alongasubsequence,weareinpositiontoapplyCorollary5.4.24to(asubsequenceof)thefollowingsequence:rkrku1,k(z)=uk(z)+2(α+1)log,z∈B1.22Thusweobtainu1,k(0)=maxu1,k+O(1),ask→+∞;|z|≤1thatis,uk(0)=maxuk+O(1),ask→+∞.|z|≤1r2kOntheotherhand,from(5.5.65)weseethatzk∈B1(0)forlargek,andweconclude2rkuk(0)≥uk(zk)−C,forasuitableconstantC>0.Butthisisimpossible,sincetheestimateabovetogetherwith(5.5.63)leadstoacontradictionofTheorem5.5.39.Remark5.5.43Concerningthe“inf+sup”estimateof(5.5.46),afirst(weaker)ver-sionwasestablishedbyShafririn[Sh]underthesoleassumption:00,asimilaruseoftheLiouvilleformula(asworkedoutin[BT1])onlyenablesonetoderive(5.5.44).Wegivean5indicationofthisfactforα∈(0,+∞)N{−1+N}.Inthiscase,aclassification2resultin[BT1]assertsthatallsolutionsfor−u=|z|2αeuinB1takeoneofthefollowingforms:8|(1+α)ψ(z)+zψ(z)|2u(z)=log,(5.5.66)(1+|z|2(α+1)|ψ(z)|2)2or8|(1+α)ψ(z)−zψ(z)|2u(z)=log,(5.5.67)(|z|2(α+1)+|ψ(z)|2)2withψholomorphicinB1satisfyingψ(0)=0and(1+α)ψ(z)±zψ(z)=0inB1wherethe±signischosenaccordingtowhether(5.5.66)or(5.5.67)isconsidered.Thus,following[Sh],wedefine8|(1+α)ψ(z)±zψ(z)|2v(z)=log,z∈B1,(1+|ψ(z)|2)2whereagainthe±signischosenaccordingtowhetherweuse(5.5.66)or(5.5.67).SincevissuperharmonicinB1,wefind8(1+α)2|ψ(0)|2log=v(0)≥minv=minv=minu.(1+|ψ(0)|2)2B1∂B1B1
Ontheotherhand,ifusatisfies(5.5.66),thenu(0)=log8(1+α)2|ψ(0)|2andweconclude:44
64(1+α)|ψ(0)|4u(0)+infu≤log≤log64(1+α).∂B1(1+|ψ(0)|2)28(1+α)2Whereas,ifusatisfies(5.5.67),thenu(0)=log2and|ψ(0)|2
64(1+α)2u(0)+infu≤log≤log64(1+α).∂B1(1+|ψ(0)|2)2Ineithercase,(5.5.44)isestablished. 5.5Aquantizationpropertyintheconcentrationphenomenon2135.5.4AQuantizationpropertyThegoalofthissectionistoestablishthefollowingresult:Theorem5.5.44Letuksatisfy(5.5.1),(5.5.2)andassume(5.5.6),(5.5.7)hold.Then(5.5.4)alsoholds.Westartwithsomepreliminaries.Firstnoticethat(5.5.1)and(5.5.2)implythat!|z|2αkVuk∀r∈(0,1),∃Cr>0:ke≤Cr,(5.5.68)Brzeroistheonlyblow-uppointforukinB1.(5.5.69)Asalreadymentioned,Theorem5.5.36willplayacrucialroleinprovingTheorem5.5.44asitimpliesthefollowingresult:Proposition5.5.45UndertheassumptionofTheorem5.5.36,supposefurtherthatforsome0<δk0suchthat,foreveryr∈(δk,rk),wehave:supuk≤βinfuk+2(αk+1)(β−1)logr+C.(5.5.71)|z|=r|z|=rDefineuk,r(z)=uk(rz)+2(αk+1)logr.1zWearegoingtoapplyTheorem5.5.36touk,rwithε0,r=ε0,zk,r=k,andrVk,r(z)=Vk(rz).Sincevk,r(z)=uk,r(|zk,r|z)+2(αk+1)log|zk,r|=vk(z)weconcludethatforasuitableconstantCdependingonlyonb1,b2andA|zk|(i)eithermaxvk→−∞andinfuk,r≤maxvk+2(αk+1)log+C;|z|=≤ε0B1|z|≤ε0r(ii)orvk(0)→+∞andinfuk,r≤−uk,r(0)+C.B1 2145TheAnalysisofLiouville-TypeEquationsWithSingularSourcesIncase(i),wefindinfuk≤maxvk+2(α+1)log|zk|−4(αk+1)logr+C.(5.5.72)|z|=r|z|≤ε0Hence,using(5.5.72)into(5.5.71),wederivetheestimate!|z|2αkVkeuk≤Ceβmax|z|≤ε0vk|zk|2(αk+1)β1−12(αk+1)β2(αk+1)βδr{δk<|z|0suchthat,alongasubsequence,wehave(i)eithersup{uk+2(αk+1)log|z|}≤C;(5.5.74)|z|≤2ε0(ii)orthereexistsequences{zj,k}⊂B{0},j=1,...,m,satisfying1.zj,k→0,uk(zj,k)+2(αk+1)log|zj,k|→+∞,ask→+∞,(5.5.75)∀j=1,...,m.1|z2.SetDk={z:0<|z|≤2ε0|z1,k|}∪{z:|z|≥m,k|},then2ε0sup{uk+2(αk+1)log|z|}≤C.(5.5.76)Dk3.Ifm≥2,thenforeveryj=1,...,m−1:|zj,k|→0,ask→+∞;|zj+1,k|(5.5.77)sup{uk(z)+2(αk+1)log|z|}≤C.{1|z|≤|z|≤2ε|z|}2ε0j,k0j+1,k 5.5Aquantizationpropertyintheconcentrationphenomenon215Proof.AccordinglytoProposition5.4.26,thereexistε10∈(0,)andaconstantC>02suchthat,eitheralternative(5.5.74)holdsorthereexistsasequencez1,k∈B1{0}suchthatz1,k→0,and(alongasubsequence)uk(z1,k)+2(αk+1)log|z1,k|→+∞,ask→+∞.Moreover,sup{uk(z)+2(αk+1)log|z|}≤C.{|z|≤2ε0|z1,k|}1),repeatananalogousalternativeintheset{|z|≥1|zForε∈(0,1,k|}.Bytakingε022εsmallerifnecessary,weobtaineither:sup{uk(z)+2(αk+1)log|z|}≤C,{|z|≥1|z|}∩B2ε01,k1whichwouldyieldtothedesiredstatementwithm=1,orthereexistsasequenceyk∈B1{0}suchthat,ask→+∞,|z1,k|→0,|yk|anduk(yk)+2(αk+1)log|yk|→+∞.(5.5.78)Sincezeroistheonlyblow-uppointforukinB1,bynecessity,yk→0.(5.5.79)Inthissecondalternative,wearegoingtoidentifyasecondsequencez2,k,byconsid-eringtheextremalproblem:sup{uk(z)+2(αk+1)log|z|},(5.5.80){1|z|≤|z|≤2ε|y|}2ε01,kkforε∈(0,1).2Iftheexpressionin(5.5.80)isuniformlyboundedforsomeε∈(0,1),thenwe2simplytakez2,k=yk,andadjustε0accordinglyinordertoensurethevalidityof(5.5.77)forj=1.Otherwise,weobtainanewintermediatesequenceofpointswiththesamepropertiesof(5.5.78)and(5.5.79),butinfinitesimalwithrespecttoyk.Repeatthesameanalysisof(5.5.80)butwithykreplacedbysuchnewsequence.Asbefore,itmayleadtoanewsequence,inwhichcasewecontinueinthesameway.AsintheproofofProposition5.4.26,eachofsuchnewsequencescontributesbyanamountof8πtothevalueofβ.Soafterafinitenumberofsteps,wemustobtainasequenceforwhich(5.5.80)isuniformlyboundedforsomeε∈(0,1).Thesequencewilldefine2z1)inordertoguaranteethat(5.5.77)holdswithj=1.2,k,whereweadjustε0∈(0,2Weiteratetheargumentabovebyreplacingz1,kwiththenewsequencez2,k.Atthispoint,weareeitherabletocheck(5.5.75),(5.5.76),and(5.5.77)form=2,orobtain 2165TheAnalysisofLiouville-TypeEquationsWithSingularSourcesathirdsequenceforwhichwecanverify(5.5.75)and(5.5.77)forj=1,2.Werepeattheargumentaboveforsuchnewsequence,toeitherfindthatm=3,orcontinueuntilweobtain(afterafinitenumberofsteps)them-sequencesthatallowustoverifythedesiredproperties.Alternative(i)inProposition5.5.46iseasytohandleaswehave:Proposition5.5.47IfthesequenceukinTheorem5.5.44satisfies(5.5.74),thenβ=8π(1+α).Proof.ByCorollary5.4.24,thevalidityof(5.5.74)impliesthatuk(0)=maxuk+0(1)→+∞,ask→+∞.(5.5.81)|z|≤2ε0Souk(0)−εk=e2(αk+1)→0,ask→+∞,andalongasubsequence,ξ2(R2),k(z)=uk(εkz)+2(αk+1)logεk→ξuniformlyinClocwithξasdefinedin(5.5.21)–(5.5.22).Since!|z|2αeξ=8π(1+α),R2wefindRk→+∞suchthat,alongasubsequence,!|z|2αVukke→8π(1+α),ask→+∞.{|z|≤Rkεk}Foreveryr∈(0,ε0),wecanuseProposition5.2.10toobtaintheestimatesupuk≤βinfuk+2(αk+1)(β−1)logr+C,(5.5.82){|z|=r}{|z|=r}withβ∈(0,1)andC>0independentofkandr.Furthermore,wecanapplyTheorem5.5.39touk,r(z)=uk(rz)+2(αk+1)logrtofindinfuk≤−uk(0)−4(αk+1)logr+C,(5.5.83)|z|=rwhich,combinedwith(5.5.82),yieldstotheestimateCe−βuk(0)|z|2αkVukke≤,(5.5.84)r2(αk+1)β+1for|z|=randCindependentofrandk. 5.5Aquantizationpropertyintheconcentrationphenomenon217Consequently,by(5.5.84)wederive!|z|2αkVuk−βuk(0)11ke≤Ce−(Rkεk)2(αk+1)βε2(αk+1)β{εkRk≤|z|≤ε0}0C≤→0,ask→+∞.2(αk+1)βRkSo!!|z|2αkVkeuk=|z|2αkVkeuk+o(1)=8π(1+α)+o(1),{|z|≤ε0}{|z|≤εkRk}andthedesiredconclusionfollowsbylettingk→+∞.AlastingredientneededfortheproofofTheorem5.5.44isthefollowingresult:Lemma5.5.48Letuksatisfy(5.5.1),withαk=0andVksatisfying(5.5.7).Supposethat(5.5.2)holdswithβ<16π.Forr0∈(0,1),letzk∈B¯rsuchthatuk(zk)=0maxuk(zk).ThenB¯r0max{uk(z)+2log|z−zk|}0.(5.5.88)kek→+∞{|z−˜zk|<δ|˜zk|}Ontheotherhand,settingu˜k(0)−εk=e2→0,k→+∞, 2185TheAnalysisofLiouville-TypeEquationsWithSingularSourcesandinviewof(5.5.86),wecanapplyLemma5.4.21toξk(z)=uk(εkz)+2logεkandseethatξksatisfies(5.5.26).Intermsofu˜k,thisimpliesthefollowing:!∀ε>0,∃Rε>0:V˜u˜k≥8π−ε,(5.5.89)ke{|z|≤Rεεk}forlargek∈N.Furthermore,from(5.5.86)and(5.5.87),wederiveu˜k(0)+2log|˜zk|≥˜uk(z˜k)+2log|˜zk|→+∞;thatis,εk→+∞,ask→+∞.|˜zk|Therefore,theset{|z−˜zk|<δ|˜zk|}∩BRεisemptyforanyδ∈(0,1)andR>1,kprovidedthatkissufficientlylarge.Butthisisimpossible,since(5.5.88)and(5.5.89)wouldimplythatβ≥16π−εforeveryε>0andthuscontradictourassumptionofβ<16π.Hence,weconcludethatu˜ksatisfies(5.5.74)(withαk=0).Consequently,(5.5.85)holds,andwecanapplyProposition5.5.47tou˜k,toobtainβ=8π.
Wearefinallyreadytogive:ProofofTheorem5.5.44.InviewofProposition5.5.47,weonlyneedtoconsiderthecasewherealternative(ii)holdsinProposition5.5.46.Inthissituation,wecanapplyProposition5.5.45andderive!|z|2αkVukke→0,ask→+∞,{ε0|zm,k|≤|z|≤1}andform≥2,!|z|2αkVukke→0,ask→+∞;∀j=1,...,m−1.{1|z|≤|z|≤ε|z|}ε0j,k0j+1,kConsequently,!m!β=|z|2αkVkeuk+|z|2αkVkeuk+o(1),(5.5.90){|z|≤ε0|z1,k|}j=1{ε|z|≤|z|≤1|z|}0j,kε0j,kask→+∞.Set1D0={z:ε0<|z|<},ε0anddefinevj,k(z)=uk(|zj,k|z)+2(αk+1)log|zj,k|,z∈D0,(5.5.91)forj=1,...,m. 5.5Aquantizationpropertyintheconcentrationphenomenon219Then− vvj,kj,k=Vj,k(z)einD0,(5.5.92)!Vvj,kj,k(z)e≤C0,(5.5.93)D0withC0>0asuitableconstant.AlsoVj,k(z)=|z|2αkVk(|zj,k|z)satisfies00andsequencesvj,kj=1,...,msuchthatinD10={ε0<|z|<}theysatisfy(5.5.92),(5.5.93),(5.5.94),(5.5.99),andβ=ε0mβ0+βjwithβ0=0or8πandβjdefinedby(5.5.96).j=1Noticethatif(II)holds,thenwecanuseProposition5.4.32(c)toapplythealter-nativeabove(attheblow-uppointzero)tothefollowingsequence:ju(z)=vj,k(z0+r0z)+2logr0,z∈B1(5.5.102)k(possiblyalongasubsequence),foreveryz0∈Sj,j=1,...,mandr0>0suffi-cientlysmall.Infact,eachtime(II)holds,wecanconstructnewsequencestowhichwecanapplyagaintheabovealternative.Ontheotherhand,eachtimethatalternative(II)holdsforanyofsuchsequencesitcontributesbyanamountofatleast16πtothevalueβin(5.5.100).Therefore,afterafinitenumberofsteps,wecannolongersupposethevalidityof(II),andweendupwithfinitelymanysequencesforwhich(I)holds.Thisprovestheclaim.Atthispointwecancompletetheproofof(5.5.4),justbyapplyingtheclaimtoj(asubsequenceof)udefinedin(5.5.102)withvj,kin(5.5.91)andanyblow-uppointkz0∈Sj.Consequently,wefindthatβj∈8πN,andwederive(5.5.4)bytakingintoaccountProposition5.5.71togetherwith(5.5.97)and(5.5.98).5.5.5ExamplesToillustratethecontent(andsharpness)ofTheorem5.5.44,inthissectionwepresentsomeinstructiveexamples.zα+1φ(z)Firstifwetakef(z)=inLiouville’sformula(2.2.3),withφandψλψ(z)holomorphicfunctionsthatarenon-vanishingattheoriginandwithλ∈R,then 5.5Aquantizationpropertyintheconcentrationphenomenon2218λ2|(α+1)φ(z)ψ(z)+z(φ(z)ψ(z)−φ(z)ψ(z))|2uλ(z)=log2(5.5.103)λ2|ψ(z)|2+|φ(z)|2|z|2(α+1)definesasolutionfor−u=|z|2αeu,(5.5.104)inadomainDwhere(α+1)φ(z)ψ(z)+zφ(z)ψ(z)−φ(z)ψ(z)nevervanishes.Bysuitablechoicesofψ,φandλ,weareabletoconstructsolutionsequencesukfor(5.5.104)satisfying2αeuk8πmδ|z|z=0,weaklyinthesenseofmeasureinB1,(5.5.105)foranygivenm∈N;or2αeuk(8π(1+α)+8πm)δ|z|z=0,weaklyinthesenseofmeasureinB1,(5.5.106)foranygivenm∈N∪{0}.OurmethodisinspiredbytheconstructiongivenbyX.X.Chenin[Chn]toobtain(5.5.105)incaseα=0.Westartwith(5.5.105).In(5.5.103),takeφ(z)=1,ψ(z)=(zm−1)eg(z),(5.5.107)withgholomorphicinB1andg(0)=0suchthat
zmlogm(α+1)(zm−1)−mzm−z(zm−1)g(z)=−(α+1)eα+1(5.5.108)Namely,g(z)intheholomorphicfunctionoverCdefinedbytheconditions:⎧
⎨zmlogm(α+1)eα+1−mzm+(α+1)(zm−1)g(z)=z(zm−1),(5.5.109)⎩g(0)=0.Noticethattheright-handsideof(5.5.109)isalsowell-definedatz=0andatthem-complexrootsofunity:2πjizj=em,j=0,1...,m−1.(5.5.110)Consequently,foreveryλ∈R,⎛
⎞zmlogm8(α+1)2λ2|eg(z)|2|eα+1|2vλ(z)=log⎝2⎠(5.5.111)|z|2(α+1)+λ2|zm−1|2|eg(z)|2definesasolutionfor(5.5.104)inthewholecomplexplane.Ournexttaskistodeter-mineasequenceλk→+∞suchthat!2αvλ|z|ek→8πm,ask→+∞.(5.5.112){|z|0sufficientlysmallsothattheballsBδ(zj)areεmutuallydisjointforeveryj=0,1,...,m−1,andthefollowinghold:(1−ε)|eg(zj)|2<|eg(z)|2<(1+ε)|eg(zj)|2,(1−ε)2<|z|2(α+1)<(1+ε)2,
2zmlogm(1−ε)3m2<(α+1)eα+1<(1+ε)3m2,m2(1−ε)m2<z−1<(1+ε)m2,z−zjforeveryz∈Bδε(zj),and∀j=0,1,...,m−1.Setσj=meg(zj)andrj,ε=δεσj.Byvirtueofthoseestimates,wefind4!!1−ε12αvλ(z)8≤|z|e1+ε(1+|z|2)2{|z|0,thereexistsδε>0andλε>1suchthat∀λ>λε,!|z|2αevλ(z)=8π+O(ε),∀j=0,1,...,m−1.Bδε(zj)m5−1Ontheotherhand,inR,δ=BRBδ(zj),R>1,wehave:j=0
!Rmlogm2αv8π(α+1)2R2(α+1)eα+1λ|z|e≤.(5.5.114)λ2δ4mmin|eg(z)|2R,δ|z|≤R1andR=k,wefindδHence,bychoosingε=k→0andλk→+∞suchthat,askk→+∞,wehave!2αvδ|z|ek=8πm+o(1),m5−1Bδk(zj)j=1 5.5Aquantizationpropertyintheconcentrationphenomenon223!|z|2αevλ=o(1).m5−1BkBδk(zj)j=1Inparticularnoticethat,from(5.5.114),bynecessity:λk→+∞.(5.5.115)kα+1InB1define,uk(z)=vλ(kz)+2(α+1)logk;kthatis,
2(kz)mlogm8(α+1)2k2(α+1)λ2|eg(kz)|2eα+1kuk(z)=log.(k2(α+1)|z|2(α+1)+λ2|(kz)m−1|2|eg(kz)|2)2kHence,uksatisfies(5.5.106)inB1andbasedonourchoiceofλk:
u8k2(α+1)λ2σ2k(zj/k)=logkj→+∞,∀j=0,1,...,m−1,!!|z|2αeuk=|z|2αevλk=8πm+o(1),B1Bksup|z|2αeuk→0,foreveryr∈(0,1),r≤|z|≤1ask→+∞.Thus,ukverifies(5.5.105).Remark5.5.49Observethat,althoughzeroisablow-uppointforuk,k2(α+1)u2k(0)=log8(α+1)→−∞,ask→+∞,λ2kasfollowsfrom(5.5.115).Inordertoconstructasequencesatisfying(5.5.106),weproceedinananalogousway.Form=0,justtakeλk→+∞,andlet8(α+1)2λ2uk.k(z)=log(1+λ2|z|2(α+1))2kThisfunctionsatisfies(5.5.104)inB1,inadditiontothefollowingproperties:(i)uk(0)=log8(α+1)2λ2→+∞,ask→+∞;k'2αu'8(α+1)2|z|2αλ22'|z|2α(ii)|z|ek=k=8(1+α);(1+λ2|z|2(α+1))2(1+|z|2(α+1))2{|z|≤1}{|z|≤1}k{|z|≤λ1/1+α}k2αuk1(iii)sup|z|e=O,foreveryr∈(0,1).{r≤|z|≤1}λk 2245TheAnalysisofLiouville-TypeEquationsWithSingularSourcesSince!2α!2α|z||z|π→=,ask→+∞,(1+|z|2(α+1))2(1+|z|2(α+1))2α+1{|z|≤λ1/1+α}R2kandinviewofthepropertiesabove,wepromptlyverifythatuksatisfies(5.5.106)withm=0.Form∈N,in(5.5.103)take1mg(z)ψ(z)=andφ(z)=λ(z−1)e,λwithg(z)theholomorphicfunctiondefinedbytheconditions:⎧
⎨zmlogm+iπmmg(z)=−(α+1)eα+1+mz+(α+1)(z−1),z(zm−1)⎩g(0)=0.Hence,
zmlogm+iπ8(α+1)2λ2|eg(z)|2|eα+1|2vλ(z)=log(1+λ2|z|2(α+1)|zm−1|2|eg(z)|2)2satisfies(5.5.104)inthecomplexplane.Similarlytothecase(5.5.111),wecanestablishthat∀ε>0thereexistsδε>0smallandλε>1:!|z|2αevλ=8πm+O(ε),(5.5.116)m5−1Bδε(zj)j=0foreveryλ≥λε,withzjdefinedin(5.5.110).Moreover,forδ>0small,setDδ=m5−1Bδ(zj)∪Bδ(0).Wethenhavej=0!22(α+1)Rmlogm2αv8π(1+α)Reα+1λ|z|e≤.(5.5.117)λ2δ4(m+1)min|eg(z)|2BRDδ|z|≤ROntheotherhand,aroundtheorigin,weseethatforanygivenε>0thereexistsδε>0suchthatforeveryz∈Bδ(0),ε
2g(z)+zmlogm+iπ1−ε<eα+1<1+ε,2mg(z)1−ε<(z−1)e<1+ε. 5.6Theeffectofboundaryconditions225Consequently,inBδ(0)thefollowingestimateholds:ε8(α+1)2(1−ε)λ2|z|2α2αv8(1+α)2(1+ε)λ2|z|2αλ≤|z|e≤.(1+(1+ε)λ2|z|2(α+1))2(1+(1−ε)λ2|z|2(α+1))21Lettingrε±=δε(1±ε)2(1+α),wefind!22α2λ|z|8(α+1)(1+(1±ε)λ2|z|2(α+1))2Bδ(0)ε!2α2|z|=8(α+1)→8π(1+α),asλ→+∞.(1+|z|2(α+1))2{|z|≤r±λ1/1+α}εThus,forδε>0sufficientlysmallandλε>1sufficientlylarge,wecanalsoensurethat!|z|2αevλ=8π(1+α)+O(ε),∀λ≥λε.(5.5.118)Bδε(0)Atthispoint,wecancombine(5.5.116),(5.5.117),and(5.5.118)tofindasequenceλk→+∞,suchthat!2αvλ1|z|ek=8π(1+α)+8πm+O(),ask→+∞.k{|z|≤k}Thus,exactlyasabove,weseethatuk(z)=vλ(kz)+2(α+1)logkkverifies(5.5.106).5.6Theeffectofboundaryconditions5.6.1PreliminariesInthisSectionwearegoingtodiscussthe(strong)effectthattheboundarycondition(5.4.36)impliesontheblow-upbehaviorforasolutionsequenceof(5.5.1),(5.5.2).Firstofall,wenoticethatacomparisonbetweenTheorem5.4.28andProposition5.5.47suggestsaconnectionbetweentheconditionsmaxuk−minuk≤C(5.6.1)∂B1∂B1andsup{uk(z)+2(αk+1)log|z|}≤C(5.6.2){|z|0satisfy8(N+1)zN+1→+∞,ask→+∞.k→0,λk|zk|Indeed,byvirtueof(2.2.13)and(2.2.15),weseethatuksatisfies(5.5.1)withVk=1andαk=N,aswellas,(5.5.2)andtheboundarycondition(5.6.1).Incontrast,(5.6.2)failsforeveryr∈(0,1).Wethushave:zN+1k→0,uk(zk)+2(N+1)log|zk|=2logλk|zk|→+∞,k→+∞.ItisalsointerestingtonotethatukblowsupalongtheN+1sequences:zj,k=2πjizkeN+1,j=0,1,2...,N.Moreprecisely,settingvk(z)=uk(|zk|z)+2(N+1)log|zk|,andzkp0=lim,k→+∞|zk|(takeasubsequenceifnecessary),weobtainN2πj|z|2Nevk8πδp,withpj=eN+1ip0,jj=0weaklyinthesenseofmeasure.Weshallseethatsucha“multi-peak”profilecannotoccurwhenα>0isnotaninteger(seeCorollary5.6.56).Infact,tostrenghtenevenfurthertheconnectionbetween(5.6.1)and(5.6.2)inSection5.6.3below,weshallprovethatthestrong“inf+sup”estimate(5.5.61)holdsforfunctionssubjecttotheboundarycondition(5.6.1).SeeTheorem5.6.59belowandcompareitwithCorollary5.5.42. 5.6Theeffectofboundaryconditions2275.6.2Pointwiseestimatesoftheblow-upprofileThegoalofthissectionistoprovidepointwiseestimatesforsolutionsequencesuk∈C2(B1)∩C0(B¯1)satisfying⎧⎪⎪−uk=|z|2αkVkeukinB1,⎨maxuk−minuk≤c0,(5.6.4)⎪⎪∂B1∂B1⎩|z|2αkVkeuk→βδz=0,weaklyinthesenseofmeasureinB¯1withc0>0asuitableconstant.FollowingtheapproachofBartolucci–Chen–Lin–Tarantello[BCLT],wehave:Theorem5.6.51Letuksatisfies(5.6.4)andassumethat(5.5.6)and(5.5.7)hold.Ifuk(0)=maxuk+O(1),(5.6.5)B¯1thenalongasubsequence,wehaveeuk(0)uk(z)−log2≤C,(5.6.6)1+γαVk(0)euk(0)|z|2(αk+1)∀z∈B¯1andasuitableconstantC>0.1,withγα=28(α+1)Observethat,whenα∈(0,+∞)N,then(5.6.5)holdsautomatically(seeCorol-lary5.6.56).Ontheotherhand,whenα∈N,example(5.6.3)showsthatassumption(5.6.5)isnecessaryforthevalidityof(5.6.6).InordertoestablishTheorem5.6.51,westartwithsomepreliminaryobservations.Firstofall,bypassingtoasubsequenceifnecessary,wearegoingtosupposethat(5.5.12)and(5.5.13)hold.Setuk(0)−εk=e2(αk+1)(5.6.7)andnotethatεk→0,ask→+∞,sinceukblowsupinB1and(5.6.5)holds.Defineξk(z)=uk(εkz)+2(αk+1)logεk(5.6.8)inB¯1/ε.Thenξk(0)=0andkmaxξk=maxuk−uk(0)=O(1).(5.6.9)B¯B¯1/εk1SobyLemma4.1.2,wehavethatξ∞(R2),(5.6.10)kisuniformlyboundedinLloc 2285TheAnalysisofLiouville-TypeEquationsWithSingularSourcesandthatalongasubsequence,ξ22k→ξuniformlyinCloc(R),(5.6.11)whereλ0ξ(z)=log,(5.6.12)(1+γαλ0|zα+1−y0|2)2withγ1α=2andwithλ0≥1,y0∈Csatisfying8(1+α)2λ0=1+γαλ0|y0|2,forα∈N∪{0}(5.6.13)λ0=1,y0=0,forα∈(0,+∞)N(seeRemark5.4.22).Ourgoalistotakeadvantageoftheboundaryconditionin(5.6.5)inordertocomplete(5.6.11)withtheglobalestimate|ξk(z)−ξ(z)|≤CinB¯1/εk,forasuitableC>0.Tothispurpose,noticethatthefunctionϕk(z)=uk(z)−minuk∂B1satisfies− ϕk=|z|2αkVkeukinB1,0≤ϕk≤Con∂B1.SowecanuseGreen’srepresentationformulaforϕktofind!112αkVuk(y)ϕk(z)=log|y|k(y)e+φk(y),(5.6.14)2π|z−y|B1withφkuniformlyboundedinC0(B¯1)∩C2(B1).Consequently,loc!1|y|2αkVuk(y)ξk(z)=uk(εkz)−uk(0)=log|y|k(y)e2πB1|εkz−y|+φk(εkz)−φk(0)Hence,settingψk(z)=φk(εkz)−φk(0)(5.6.15)andafterachangeofvariable,wederive!1|y|2αkVξk(y)ξk(z)=log|y|k(εky)e+ψk(z),(5.6.16)2π|y−z|{|z|<1}εk∀z∈B1/ε.k 5.6Theeffectofboundaryconditions229Lemma5.6.52Foreveryε>0,∃Rε>1,kε∈NandCε>0suchthat,alongasubsequence(denotedthesameway),wehave:1ξk(z)≤(4(α+1)−ε)log+Cε,(5.6.17)|z|∀|z|≥2Rεand∀k≥kε.Proof.Weconsiderthesubsequenceforwhich(5.6.11)holdstogetherwith(5.5.12)and(5.5.13).ByTheorem5.4.28,weknowthatβ=8π(1+α)in(5.6.4).Set!!Mk=|z|2αkVk(z)euk(z)=|z|2αkVk(εkz)eξk,(5.6.18)B1B1/εksothatMk→8π(1+α),ask→+∞.(5.6.19)'Alsorecallthat|z|2αeξ=8π(1+α),forξthelimitingfunctionin(5.6.11).Conse-R2quently,foragivenε>0,wefindkε∈NandRε>1suchthatthefollowinghold:!2αξ2πε|z|e≥8π(1+α)−;(5.6.20)5(α+2)BRεandfork≥kε,2πε|Mk−8π(1+α)|<,(5.6.21)5(α+2)!!|z|2αkVξk2αξ2πεk(εkz)e≥|z|e−5(α+2)BRεBRε2ε≥2π4(1+α)−.(5.6.22)5(α+2)Inparticular,!|z|2αkVξk6πεk(εkz)e<,∀k≥kε.(5.6.23)5(α+2)B1/εkBRεToobtain(5.6.17),weuseexpressions(5.6.16)and(5.6.22)sothatforz:|z|>2Rεand∀k≥kε,wecanestimate!!1log|y||y|2αkVk(εky)eξk(y)≤1log2Rε|y|2αkVk(εky)eξk(y)2π|z−y|2π|z|BRεBRε2ε2Rε≤4(1+α)−log;5(α+2)|z| 2305TheAnalysisofLiouville-TypeEquationsWithSingularSourcesandwecanderive!1|y|2αkVξklog|y|k(εky)e2π|z−y|{|y|<|z|}2!≤1|y|2αkVξklog|y|k(εky)e2π|z−y|BRε!+1|y|2αkVξk(y)(5.6.24)log|y|k(εky)e2π|z−y|{R|z|ε<|y|<}!2≤1|y|2αkVξk2ε2Rεlog|y|k(εky)e≤4(1+α)−log2π|z−y|5(α+2)|z|BRε2ε1≤4(1+α)−log+C1,ε,5(α+2)|z|forasuitableconstantC1,ε.Furthermore,wecanuse(5.6.9)and(5.6.23)toestimate:!|y|2αkVξk(y)log|y|k(εky)e|z−y|{|z−y|<|z|}2!=(log|y|)|y|2αkVξk(y)k(εky)e{|z−y|<|z|}2!+12αkVξk(y)log|y|k(εky)e|z−y|{|z−y|<1}|z|α+1!+12αkVξk(y)log|y|k(εky)e|z−y|{1≤|y−z|<|z|}|z|α+122αk!11≤C|z|+logdy|z|α+1|z−y|{|z−y|<1}|z|α+1
!+(α+2)log|z|+O(1)|y|2αkVξk(y)k(εky)e{|y|≥|z|}212(1+α−αk)3≤2πC+εlog|z|+C2,ε.Rε5Thereforebytakingkε∈NandRεlargerasnecessary,wecanensurethatas∀k≥kεandforeveryzsuchthat|z|≥2Rε,wehave 5.6Theeffectofboundaryconditions231!1|y|2αkVξk(y)4εlog|y|k(εky)e0.Claim:For|z|≥3,+|y||z|1log≤2|log|y||+log+log2(5.6.34)|y−z||y−z|inB1/ε{z},whereasusual,f+=max{f,0}denotesthepositivepartofagivenkfunctionf.|z|Toverify(5.6.34),firstlety∈B|z|(0).Then|y−z|≥>1andso2
2+log1=0.Furthermore,|y−z||y||z|zylog=log|y|+log−≤log|y|+log2|y−z||z||z|and(5.6.34)isverifiedinthiscase.|z||z||y||z|Nowtakey:|y−z|<.Hence|y|≥>1andlog≥log2|y|≥22|y−z|log|z|>0.Therefore,+|y||z|11log=log|y||z|+log≤2log|y|+log2+log|y−z||y−z||z−y|+1=2|log|y||+log2+log.|z−y|

+Finally,ify∈B11/εB|z|(0)∪B|z|(z),thenlog=0whilek|z−y|22|y||z||y||z||z|log≥log≥log>0.|y−z||y|+|z|3 5.6Theeffectofboundaryconditions233|y||z||y||z|Consequently,log=log≤log2|y|and(5.6.34)followsalsointhiscase.|y−z||y−z|Bymeansof(5.6.34)wecanimmediatelyget(5.6.29)inviewof(5.6.31).Andweobtain!Mk1|y||z|2αkξkξk(z)+log|z|≤log|y|Vk(εky)e+O(1)2π2π|z−y|B1/εk!1|log|y|||y|2αkVξkMk≤k(εky)e+log2π2πB1/εk⎛⎞!⎜1⎟+C⎝log⎠+O(1)=O(1).|z−y|{|z−y|<1}Toobtain(5.6.30),wetakeadvantageof(5.6.29)justestablishedtodeduce:Mkξk(z)≤−log(1+|z|)+C,∀z∈B1/ε.(5.6.35)k2πSince!Mkz1z−yz2αkξk(y)∇ξk(z)+2π|z|2≤2π|z−y|2−|z|2|y|Vk(εky)e+O(εk)(5.6.36)B1/εkinBr0/εk,thenbytakingksufficientlylarge,wecanguaranteethatMk3−2(αk+1)≥.(5.6.37)2π2Weproceedtoestimatetheintegralin(5.6.36)inthethreeregionsD1=B|z|(0),D2=2B|z|(z)andB=B1/ε(D1∪D2).kk2Using(5.6.35)and(5.6.37),forklarge,wehave!!z−yz2αkξk(y)|y|2αkξk(y)|z−y|2−|z|2|y|Vk(εky)e=|y−z||z||y|Vk(εky)eD1D1!|!z|/2(5.6.38)C|y|1+2αkCdrC≤≤≤.|z|2(1+|y|)Mk/2π|z|2(1+r)3/2|z|2{|y|<|z|/2}0SinceD|z|32⊂{≤|y|≤|z|},forsufficientlylargekand|z|≥1,wefind:22!!z−yz2αkξk(y)12αk−Mk−|y|Vk(εky)e≤C|y|2πD|z−y|2|z|2D|y−z|2!22α−Mk12α+1−MkC(5.6.39)≤C|z|k2π≤C|z|k2π≤.|y−z||z|5/2{|z−y|<|z|}2 2345TheAnalysisofLiouville-TypeEquationsWithSingularSourcesFinally,notingthatz−yz|y|4|z−y|2−|z|2=|z−y||z|≤|z|,∀y∈Bk,(5.6.40)weconclude!!z−yz2αkξk(y)C2αk−MkC−|y|Vk(εky)e≤|y|2π≤(5.6.41)|z−y|2|z|2|z||z|5/2B{|y|≥|z|}k2(provided)kissufficientlylargeand|z|≥1.Theestimate(5.6.30)followsimmedi-atelyfrom(5.6.36),(5.6.38),(5.6.39),and(5.6.41).
Observethattheestimates(5.6.29)and(5.6.30)canberecastedintermsofuk,respectively,asfollows:MkMk1uk(z)=−log|z|+2(1+αk)−log+O(1),(5.6.42)2π2πεkfor3εk≤|z|≤1,andksufficientlylarge;Mkzεk∇uk(z)=−+O(5.6.43)2π|z|2|z|2for3εk≤|z|≤r0,wherer0∈(0,1)andkissufficientlylarge.Estimates(5.6.42)and(5.6.43)arecrucialtoestablishthefollowingimportantproperty:Lemma5.6.55Alongasubsequencethefollowingholds,
Mk=O(log1/ε−1+4(αk+1)k),ask→+∞.(5.6.44)2πProof.Toobtain(5.6.44),wearegoingtousePohozaev’sidentity(5.2.16)intheballB˜1}.Thisimpliesthatk={|y|≤εklogεk!62-!r∂uk−1|∇u|2r1+2αVukkdσ+ke∂B˜k∂ν2∂B˜k!
=2|y|2αkVkeuk+y·∇|y|2αkVkeuk(5.6.45)B˜k!!=2(1+αk)|y|2αkVkeuk+|y|2αk(y·∇Vuk,k)eB˜B˜kkforr=|y|. 5.6Theeffectofboundaryconditions235On∂B˜kwecanuse(5.6.43)toderive:!62-!62 -∂uk121Mk11r−|∇uk|dσ=r+Odσ∂ν222πr2ε2(log1)3kεk∂B˜k∂B˜k(5.6.46)2−1Mk1=+Olog.4πεkWhile(5.6.42)impliesthat⎛⎞!2(α+1)−Mk1k2πr1+2αkVkeukdσ=O⎝log⎠,εk∂B˜k!!1Mk|y|2αkVkeuk=Mk+|y|2αkVkeuk=Mk+O(log)2(αk+1)−2π,εkB˜kB1B˜kand!2αu12(α+1)−Mk|y|ky·∇Vkek≤O(log)k2π.εkB˜kTherefore,inviewof(5.6.37),wecanusetheestimatesabovetoconclude2Mk1−11−1+O(log)=2(αk+1)Mk+O(log)4πεkεkprovidedthatkissufficientlylarge.Thus(5.6.44)followsimmediately.ProofofTheorem5.6.51.Using(5.6.11)and(5.6.12)togetherwith(5.6.29)and(5.6.44)issufficientforconcluding:1ξk(z)−log≤CinB¯1/ε.(1+γαVk(0)|z|2(αk+1))2kConsequently,(5.6.6)followsbymeansof(5.6.7)and(5.6.8).
AsadirectconsequencesofTheorem5.6.51,weobtain:Corollary5.6.56Letuksatisfy(5.6.4)withαk→α∈(0,+∞)NandVkasin(5.5.6).Thenthesequenceuksatisfiesthepointwiseestimate(5.6.6)inB¯1.Proof.ItsufficestouseProposition5.6.50togetherwithCorollary5.4.24toverify(5.6.5).ThenbyrecallingRemark5.6.53,wecanguaranteethevalidityof(5.6.6)forthewholesequenceuk.
2365TheAnalysisofLiouville-TypeEquationsWithSingularSourcesCorollary5.6.57Letuksatisfy(5.6.4)withαk=0andVkasin(5.5.6).Foreveryr∈(0,1),thereexistsCr>0suchthatλkuk(z)−log≤Cr,inB¯r,(5.6.47)(1+Vk(zk)λk|z−zk|2)28wherezuk(zk)k→0:uk(zk)=maxukandλk=e.(5.6.48)B¯1Proof.Foragivenr∈(0,1),letu˜k(z)=uk(zk+rz)+2logr.Weeasilycheckthatthesequenceu˜ksatisfies(5.6.4),(5.6.5)withαk=0andV˜k(z)=Vk(zk+rz).Furthermore,given|z1|=|z2|=1,weuse(5.6.14)andfind!1|zk+rz2−y|uk(y)|u˜k(z1)−˜uk(z2)|≤logVk(y)e+O(1),2π|zk+rz1−y|B1and!|zk+rz2−y|uk(y)logVk(y)e→0,ask→+∞.|zk+rz1−y|{|y|<1}Therefore,u˜ksatisfies(5.6.1),aswellas,alltheotherassumptionsofTheorem5.6.51.Sinceu˜k(0)=maxu˜k,wecanalsouseRemark5.6.53toconcludethevalidityoftheB¯1pointwiseestimate(5.6.6)forthewholesequenceu˜k.Thatis,eu˜k(0)u˜k(z)−log
2≤CinB¯1.(5.6.49)1+V˜k(0)eu˜k(0)|z|28From(5.6.49)andthedefinitionofu˜k,weimmediatelyderive(5.6.47).
Remark5.6.58Atthispoint,itisnaturaltoconjecturethat,whenα=N∈N,thefollowingestimateshouldholdforasolutionsequenceuk,of(5.6.4):λkuk(z)−log
2≤CinB¯1,(5.6.50)1+Vk(zk)λk|zN+1−zN+1|28(N+1)2kwithzkandλkasdefinedin(5.6.48). 5.6Theeffectofboundaryconditions237Insupportof(5.6.50),noticethatincase:λN+1=O(1);(5.6.51)k|zk|thatis,|zk|−uk(zk)=O(1),withεk=e2(N+1),εkthenwecanargueasinCorollary5.4.24andverifythat(5.6.5)issatisfied.Thenwemayapply(5.6.6)togetherwith(5.6.51)toobtain(5.6.50).Thus,thetrulydelicatecasetoconsideroccurswhenzk→+∞.(5.6.52)εkInthissituation,wecanonlygiveapartialcontributiontowards(5.6.50)byproving,inthefollowingsection,thevalidityofthestrongerversionofthe“sup+inf”estimateasgivenin(5.5.61).5.6.3Theinf+supestimatesrevisedForgivenc0>0,considertheproblem:⎧⎪⎪−u=|z|2αVeuinB1,⎪⎨'|z|2αVeu≤c0,(5.6.53)⎪⎪B1⎪⎩maxu−minu≤c0.∂B1∂B1Thefollowingholds:Theorem5.6.59Ifusatisfies(5.6.53)andif(5.5.43)holds,thenforeveryr∈(0,1)thereexistsaconstantC>0dependingonlyonr,α,b1,b2,A,andc0suchthatsupu+infu≤C.(5.6.54)BrB1Theorem5.6.59shouldbecomparedwithCorollary5.5.42.Ithastheadvantageofrelyinguponassumptionsthatareoftenavailableintheapplications.Proof.Toobtain(5.6.54)wearguebycontradiction,andsimilarlytoCorollary5.5.42,weassumethatthereexistsasequenceuk,satisfying:⎧⎪⎪−uk=|z|2αVkeukinB1,⎪⎨'|z|2αVkeuk≤c0,(5.6.55)⎪⎪B1⎪⎩maxuk−minuk≤c0,∂B1∂B1 2385TheAnalysisofLiouville-TypeEquationsWithSingularSourceswithVksatisfying(5.5.7)andsuchthatforasequence{zk}⊂B1wehave:uk(zk)+infuk→+∞(5.6.56)B1and(byvirtueofCorollary5.5.7)zk→0(5.6.57)ask→+∞.Withoutlossofgenerality,wecanfurtherassumeuk(zk)=maxuk,(5.6.58)B¯1|z|2αVukke→βδz=0,(5.6.59)weaklyinthesenseofmeasureinB1.Thus,wecanapplyTheorem5.4.28andconcludethat(5.6.59)holdswithβ=8π(1+α).(5.6.60)uk(zk)−Setεk=e2(α+1).Asalreadymentionedabove,weonlyhavetoconsiderthecasewherezk→+∞,ask→+∞;(5.6.61)εkotherwisewecouldverify(5.6.5)anduse(5.6.6)tocontradict(5.6.56).Alsonoticethat(5.6.61)isequivalentto,uk(zk)+2(α+1)log|zk|→+∞,ask→+∞.(5.6.62)Toaccountfor(5.6.62),weconsidervk(z)=uk(|zk|z)+2(α+1)log|zk|definedinB2k:={z:|z|≤}.Wehave:|zk|!|z|2αVvkk(|zk|z)e=8π(1+α)+o(1),(5.6.63)Bkasaconsequenceof(5.6.59)and(5.6.60).Furthermore,(5.6.62)impliesthat,zkvk→+∞,ask→+∞;|zk|thatis(alongasubsequence),vkadmitsablow-uppointz0intheunitcircle. 5.6Theeffectofboundaryconditions239Claim:Ifz0isablow-uppointforvk,thenthereexistsρ0>0sufficientlysmallandC>0suchthatmaxvk−minvk≤C.(5.6.64)∂Bρ0(z0)∂Bρ0(z0)ToestablishtheclaimweuseGreen’srepresentationformulaasin(5.6.14)andforrk=|zk|wewrite!1|˜x−y|2αvkvk(x)−vk(x˜)=log|y|Vk(rky)e+O(1),(5.6.65)2π|x−y|Bk∀x,x˜∈k.Letρ0∈(0,1)sufficientlysmall,sothatz0istheonlyblow-uppointfor(zρ0vkinB2ρ0)⊂BkandletD0={y∈Bk:≤|y−z0|≤2ρ0}.Itiseasytocheck02thatforx,x˜∈∂Bρ(z0)0|˜x−y|suplog≤C1,(5.6.66)BkD0|x−y|whilesupvk≤C2,(5.6.67)D0forsuitablepositiveconstantsC1andC2.Thus,foreveryx,x˜∈∂Bρ(z0),from0(5.6.65)wecanestimate:!1|˜x−y|2αvk|vk(x)−vk(x˜)|≤log|y|Vk(rky)e2π|x−y|D0!1|˜x−y|2αvk+log|y|Vk(rky)e2π|x−y|kD0!!+O(1)≤C2αvk3|log|z||+C4|y|Vk(rkz)e+O(1)≤CB3ρ0(0)kandtheClaimfollows.Thus,wecanuseCorollary5.6.57forvzkandderivekaround|zk|µkvk(z)−log≤C(5.6.68)(1+1Vk(zk)µk|z−zk|2)28|zk|
zk(zvk|zk|inBρ0),withµk=e.0Asaconsequenceof(5.6.68)wefind!|z|2αVvkk(|zk|z)e=8π+o(1),ask→+∞,(5.6.69)Bδ(z0) 2405TheAnalysisofLiouville-TypeEquationsWithSingularSources∀δ∈(0,ρ0),andzkvk+infvk≤C(5.6.70)|zk|∂Bρ0(z0)(notethat(5.6.69)alsofollowsbyTheorem5.4.28appliedwithα=0).By(5.6.63)and(5.6.69),weseethatzerocannotbeablow-uppointforvk.Indeed,ifthiswasthecase,thenwecouldusetheClaimabovewithz0=0,andbyTheorem5.4.28concludethat!|z|2αVvkk(|zk|z)e=8π(1+α)+o(1),Bε(0)foreveryε>0sufficientlysmall,incontradictionwith(5.6.63)and(5.6.69).Henceforsmallε0>0,necessarily:maxvk→−∞,ask→+∞.(5.6.71)B¯2ε0Thus,wereadilycheckthatalternative(a)ofTheorem5.5.36appliestovkandimplies:infuk≤maxvk+2(α+1)log|zk|+C.(5.6.72)B1{|z|≤ε0}Furthermore,using(5.6.71)and(5.6.65),wecanargue,asintheproofoftheClaimabove,tofindaconstantC>0suchthat|vk(x)−vk(x˜)|≤C,∀x∈Bεandx˜∈∂Bρ(z0).00Inotherwords,maxvk≤minvk+C.(5.6.73)Bε∂Bρ0(z0)0Consequently,from(5.6.72)and(5.6.73),wededucethefollowingestimatezkzkuk(zk)+infuk≤maxvk+vk+C≤minvk+vk+C.B1B¯ε0|zk|∂Bρ0(z0)|zk|Therefore,wecanuse(5.6.70)toobtainacontradictionto(5.6.56),andconcludetheproof.Asamatteroffact,theargumentsaboveallowustodeducea(global)“inf+sup”estimateinthesamespiritofCorollary5.5.40,relativetosolutionsoftheproblem:⎧⎪⎪−u=W(z)euin,⎨'W(z)eu≤c0,(5.6.74)⎪⎪⎩sup−inf≤c0,∂∂ 5.7Theconcentration-compactnesssprinciplecompleted241forgivenc0>0,whereweassumethat$mW(z)=|z−z2αii|V(z)(5.6.75)i=1with{z1,...,zm}⊂distinctpointsandαi≥0foreveryi=1,...,m,(5.6.76)and0,1()satisfies00sufficientlysmall,weconsiderthesequenceu˜k(z)=uk(q+r0z)+2(αk+1)logr0,definedforz∈B1,wherewetakeαk=0ifq∈{/z1,...,zm}orαk=αi,kifq=ziforsomei=1,...,m.Hence(5.7.1)followssimplybyapplyingTheorem5.5.44tou˜k.Ifcase(5.7.3)alsoholds,thenaccordingto(5.6.64)wecanassume:supuk−infuk≤C,∂Br(q)∂Br0(q)0forasuitableC>0.Therefore,inthiscase,(5.7.3)followsbyapplyingTheorem5.4.28tothesequenceu˜k.Wegobacktoanalyzingouroriginalproblem(5.1.1),whereweconsidersolution-sequencesofLiouvilleequationsintheMean–Fieldform.Infact,wediscussamoregeneralframework,whereukisdefinedoveraclosedsurface(M,g).Denotewithg,∇ganddσgrespectivelytheLaplace–Beltramioper-ator,thegradientandthevolumeelementrelativetothemetricgonM.Also,letdgdenotethedistancefunctionon(M,g).Letwksatisfy
'hk(x)ewk1−gwk=µkhk(x)ewkdσg−|M|inM,'M(5.7.4)wk∈H1(M):wkdσg=0,Mandforthemoment,weassumethatµk→µ>0(5.7.5)!u0,k∈C(M):u1hk=e0,k∈L(M)andu0,kdσg=0.(5.7.6)MWestartbyobservingthefollowingusefulfact:Lemma5.7.62Letwksatisfies(5.7.4)andassume(5.7.5)and(5.7.6)hold.ThenwkisuniformlyboundedinW1,q(M)∀1≤q<2.Proof.TheargumentisanalogoustothatofLemma4.1.2.Hence,letp,thedualexponentofq(thatis1+1=1),sothatp>2;andletusconsiderqp!1,pϕ∈W(M)suchthat||∇gϕ||Lp(M)=1,ϕdσg=0.(5.7.7)MBytheSobolevembeddingtheorem(cf.[Au]),weknowthatϕ∈C0(M)and||ϕ||L∞(M)≤c0forasuitableconstanntc0>0.Bytakingintoaccount(5.7.5),weobtain!!whk(x)ek∇gwk∇gϕ=µk'wϕ≤µk||ϕ||L∞(M)≤CMMhk(x)ekdσgM 5.7Theconcentration-compactnesssprinciplecompleted243withC>0asuitableconstant.Therefore,!q||∇gwk||Lq(M)=sup{∇gwk·∇ϕ,ϕsatisying(5.7.7)}≤C(5.7.8)Masclaimed.Clearly,wecanextendthenotionoftheblow-uppoint,p∈MforthesequencewkinM,simplybyaskingif∃{zk}⊂M:zk→pandwk(zk)→+∞,ask→+∞.(5.7.9)Remark5.7.63Ifthesequencewksatisfies(5.7.4),thenwecanusestandardellipticregularitytheory,toseethat(5.7.9)isequivalenttotheproperty:!hwkwk(zk)−logke→+∞,ask→+∞.(5.7.10)MProposition5.7.64Letwksatisfies(5.7.4)andassumethat(5.7.5)and(5.7.6)hold.Ifhk→huniformlyinMandpisablow-uppointforwksuchthath(p)>0,then'hkewkUδ(p)liminfµk'≥8π,(5.7.11)n→∞hkewkMwhereUδ(p)={q∈M:dg(p,q)<δ}.Furthermore,ifin(5.7.5)wehaveµ=8π,thenpistheonlyblow-uppointforwk,andalongasubsequencewehave:hkewk'δpweaklyinthesenseofmeasureinM,(5.7.12)hkewkM!hwksupwk−logkedσg→−∞∀D⊂⊂M{p}.(5.7.13)DMProof.InasmallneighborhoodofpinM,wedefineanisothermalcoordinatesystemy=(y1,y2)centeredattheorigin(i.e.,y(p)=0)suchthatds2=eϕ(dy2+dy2)anddσϕ12g=edy1dy2,withasmoothfunctionϕ=ϕ(y)satisfyingϕ(0)=0,∇ϕ(0)=0and− ϕ=2KeϕinBr(y)0withKtheGausscurvatureofM.Withoutambiguity,wewritewk=wk(y)toexpresswkinsuchacoordinatesystem.Define!2wµk|y|uk(y)=wk(y)−loghkekdσg−,y∈Br(0).(5.7.14)0M4|M| 2445TheAnalysisofLiouville-TypeEquationsWithSingularSourcesHenceukadmitsablow-uppointattheoriginandsatisfies−uk=WkeukinBr(0),'0u(5.7.15)Wkek≤C,Br(0)0withWk=µkhkeϕ→µheϕ:=W,uniformlyinBr(0),andwithW(0)>0.0Therefore,wecanapplyProposition5.4.20toseethat!Wukliminfke≥8π,∀δ∈(0,r0)n→∞Bδ(0)andinturndeduce(5.7.11).Furthermore,bymeansofproperty(c)ofProposi-tion5.4.32(alongasubsequence),weknowalsothat!hwksupwk−logkedσg→−∞,(5.7.16)∂Uδ(p)Mask→+∞,andforδ>0sufficientlysmall.Nextassumethatµk→8π.Thenby(5.7.11)weseethatnecessarily'hkewkdσgUδ(p)'→1ask→∞,∀δ>0,(5.7.17)hkewkdσgMorinotherwords,'hkewkdσgMUδ(p)'→0ask→∞.hkewkdσgMTherefore,ifasaboveweuseisothermalcoordinatestolocalizeourproblemaroundapointq∈M{p},weobtainthatinthiscasethesequenceuk,givenin(5.7.14),satisfies(5.7.15)withtheadditionalproperty:!Wukke→0,ask→∞,(5.7.18)Br(0)0forr+||0>0sufficientlysmall.From(5.7.14)wealsoseethat||ukL1isuniformlybounded,aswehave:!+(y)≤|wwkuk(y)|−log−hkedσg+log|M|≤|wk(y)|+log|M|.k'Sincelog−hkewk≥0,and(byLemma5.7.62),weknowthatwkisuniformlyMboundedinL1-norm.Thus,weareinapositiontoapplyProposition5.3.13toseethatu+isuniformlykboundedinBδ(0)foreveryδ∈(0,r0).Atthispoint,recalling(5.7.16),wecanuseCorollary5.2.9andconcludesupuk≤βinfuk+C,Bδ(0)Bδ(0) 5.7Theconcentration-compactnesssprinciplecompleted245forβ∈(0,1)andC>0suitableconstants.Noticethatsincewehavemadenoassumptionsaboutthesizeofthezerosetofhk,wecannotyetconcludethatinfuk→Bδ−∞ask→−∞from(5.7.18).Nevertheless,bycompactness,wecanpatchallsuch“local”informationtogethertofindthat,foreveryδ>0sufficientlysmall,!!suphwkwkwk−logkedσg≤βinfwk−loghkedσg+CMUδ(p)MMUδ(p)Mwithsuitableconstantsβ∈(0,1)andC>0.Byvirtueof(5.7.16)wecheckthat!hwkinfwk−logkedσg→−∞,MUδ(p)M(alongasubsequence)and(5.7.13)follows.Alsowederive(5.7.12)byvirtueof(5.7.17)Toanalyzethebehaviorofwkaroundageneralblow-uppoint,wefocusonthecasewhere$m(dαj,khk(q)=g(q,pj))Vk(5.7.19)j=1withp1,...,pmdistinctpointsonM,and0≤αj,k→αjask→+∞,j=1,...,m(5.7.20)V0,1k∈C(M):00.Bytakingadvantageoftheanalysiscarriedoutintheprevioussections,weobtainthefollowing“concentration/compactness”resultforwk:Theorem5.7.65Letwksatisfy(5.7.4)andassumethat(5.7.5),(5.7.6),and(5.7.19)–(5.7.21)hold.Alongasubsequence(denotedthesameway),oneofthefollowingal-ternativeholds:(i)wkconvergesuniformlyonM.(ii)Thereexistsafinitesetofblow-uppointsS={q1,...,qs}⊂Mwiththefollowingproperties:(a)thereexist{qj,k}⊂M:qj,k→qjandwj,k(qj,k)→+∞,ask→+∞,j=1,...,s;(b)!hwkmaxwk−logkedσg→−∞,∀D⊂⊂MS;(5.7.22)DM 2465TheAnalysisofLiouville-TypeEquationsWithSingularSources(c)ask→+∞,hkewksµk'→βjδq,weaklyinthesenseofmeasureonM,(5.7.23)hkewkdσgjMj=1and8πifqj∈{/p1,...,pm}βj=(5.7.24)8π(1+αi)ifqj=piforsomei∈{1,...,m}.Inparticular,setting⎧⎛⎞⎫⎨⎬=8πN∪8π⎝N+αj⎠foreverysetJ⊆{1,...,m},(5.7.25)⎩⎭j∈Jalternative(ii)thenholdsifandonlyifµin(5.7.5)satisfies:µ∈.Theorem5.7.65extendsaresultofLiin[L2]concerningthecasewherethefunctionhkisuniformlyboundedawayfromzeroinM,i.e.,inournotationwhenαj,k=0in(5.7.19)∀j=1,...,mand∀k∈N.ProofofTheorem5.7.65.Ifmaxwk≤CforasuitableC>0,thentheright-handMsideof(5.7.4)isuniformlyboundedinL∞(M).SobystandardellipticestimatesandSobolev’sembeddings(cf.[Au]),weseethatwkisuniformlyboundedinC2,γ(M)forγ∈(0,1),andwededuce(i).Henceassumethat,alongasubsequence,wehave:maxwk→+∞,ask→∞(5.7.26)Morequivalently,!hwkmaxwk−logkedσg→+∞,ask→∞.MMAsaboveforq∈M,weusealocalisothermalcoordinatesystemy=(y1,y2)cen-teredatq,suchthat,forthesequenceukin(5.7.14),wefind−uk=|y|2αkV˜keukinBr(0),'0|y|2αkV˜keuk≤C,(5.7.27)Br(0)0(forr0>0sufficientlysmall)withV˜k=µkVkeϕ;whileαk=0ifq∈{/p1,...,pm},orαk=αj,kifq=pjforsomej∈{1,...,m}.Accordingtoourassumptionsof(5.7.5)and(5.7.21),wecanapplyProposition5.4.32toeachofthelocalproblemsof(5.7.27).By(5.7.26)andacompactnessargu-ment,weconcludetheexistenceofafinitesetofblow-uppoints,S={q1,...,qs},suchthatproperties(5.7.22)and(5.7.23)holdforthesequencewk.Noticeinparticular 5.8Finalremarksandopenproblems247thatsupwk≤C,foreveryD⊂⊂MS.Withthisinformation,wecanuseellipticDestimatesasabove,tofindthat(alongasubsequence)thereholds:s2,γwk→βjG(q,qj)uniformlyinC(MS),locj=1γ∈(0,1)andGtheGreen’sfunctionin(2.5.11).Thus,wecheckthatwkadmitsuniformlyboundedoscillationsintheboundaryofasmallneighborhoodofqj∈S,foreveryj=1,...,s.Atthispoint,wecanuseTheorem5.4.28forthecorrespondinglocalizedsequenceuktoconclude(5.7.24).Remark5.7.66IfMadmitssmoothboundary∂MwhereweimposeDirichletbound-aryconditions,thenitispossibletoruleoutthatablow-uppointoccurson∂M(e.g.,see[MW]).So,forasolutionsequenceswksatisfying:wk−gwk=µk'hkeinMhkewkM(5.7.28)wk=0on∂M,theanalysisabovesufficestoobtaina“concentration-compactness”resultincompleteanalogytothatstatedinTheorem5.7.65.NoticeinparticularthatMcouldbetakenasaregularsubdomainofR2.IfwerequirethatNeumannboundaryconditionsapplyon∂M,thenwefaceamoredelicatesituationasboundaryblow-upoccursinthiscase.Wereferto[WW1],[WW2],andreferencesthereinforadiscussionofthissituation.Theanalysiscarriedoutinthischapterfollowsthespiritofthatdevelopedforsev-eralotherproblemsingeometryandphysicswhere“concentration”phenomenaoccurnaturally.Inthisframeworkoneaimstoidentifytheregimeunderwhich“concen-trations”cannotdevelop.Orwhen“concentrations”occur,oneaimstodescribe(asaccuratelyaspossible)the“bubble”profilethattheconcentrating-sequenceassumesduringblow-up.Actually,ithasbeenpossibletouseperturbationandgluingtechniquestoconstructexplicitclassesofsolutionswhichadmita“concentration”behaviorwithaprescribed“bubble”profile(asindicatedbytheblow-upanalysis).Inthisrespectandinrelationtotheaimofthesenotes,wementionthefollowingcontributions:[Re],[BC1],[BC2],[Ba],[KMPS],[NW],[Pa],[Sc1],[Sc2],[BP],[Es],[DDeM],[DeKM],[EGP],[Dr],[DHR],[AD],[Fl1],[AS],[PR],[We],andreferencestherein;andmentionthatbynomeansthisfurnishesacompletelistofreferencesonthesubject.5.8FinalremarksandopenproblemsAsusualweconcludewithadiscussionofsomeproblemsinblow-upanalysis,whichstemfromthestudyofvorticesandareleftopenbyourinvestigation.AfirstproblemconcernsthevalidityofanHarnack-typeinequalityinthefollowing“inf+sup”form: 2485TheAnalysisofLiouville-TypeEquationsWithSingularSourcesOpenproblem:Givenα0>0,Vsatisfying(5.5.43)andK⊂⊂B1compact,isittruethatforanyα∈[0,α0]asolutionoftheequation−u=|x|2αVeusatisfiessupu+infu≤CKB1withC>0aconstantdependingonlyonα0,dist(K,∂B1)andtheconstantsb1,b2,andAin(5.5.43)?Sofarweknowthattheabovepropertyholdswhenα0=0,orwhenα0>0butK⊂⊂B1{0}(cf.[BLS]),orforanyα0>0butK={0}(cf.[T6]).Thegeneralcasehasbeenhandledin[BCLT],butonlyundersomeadditionalconditionsasstatedinTheorem5.6.59andTheorem5.6.60above.Anotheraspectofouranalysiswhichremainswideopenforinvestigationconcernsthevalidityof“concentration-compactness”principlesforsystems,andmorespecifi-callyfortheSU(3)−Todasystem(4.5.3)or(4.5.11)and(4.5.35).AnyprogressinthisdirectionwouldhavedirectimpactinthestudyoftheasymptoticbehaviorofperiodicSU(3)−vorticesofthe“non-topological-type”or“mixed-type”,asgivenforinstanceinTheorem4.5.35.TowardsthisgoalwerecalltheimportantcontributionofJost–Lin–Wang[JoLW],whichtakesadvantageoftheboundaryconditionsinordertohandlethe“regular”Toda-system,wheretheDiracmeasuresareneglected.Sothetruedelicatesituation,yettobeunderstood,concernsthecasewhereDiracmeasuresareinvolvedandblow-upoccursatpointswheresuchmeasuresaresup-ported.HenceweposethefollowingOpenproblem:Determinetowhatextentthe(blow-up)analysisofsection5.4.2remainsvalidwhenthesingleequation(5.4.1)isreplaced,forinstance,byanSU(3)−Todasystemlikethefollowing:− v1,k=2|x|2α1,kV1,kev1,k−|x−xk|2α2,kV2,kev2,kinB1(5.8.1)− v2,k=2|x−xk|2α2,kV2,kev2,k−|x|2α1,kV1,kev1,kinB1,wherexk∈B1andbothV1,kandV2,ksatisfyassumptionsofthe(5.4.2)typeandα1,kandα2,ksatisfy(5.4.3).Theanalysisof(5.8.1)shouldprovideafirststeptowardsthemaingoalthatwouldbetoanswerthefollowing:Openquestion:Inwhichformdoesa“concentration-compactness”principleofthetypegivenbyTheorem5.4.34(orTheorem5.7.61),holdforasequenceofsolutionsoftheSU(3)−Todasystemof(4.5.3)(ormoregenerallyforsystemsofthe(2.5.24)type)inthepresenceofDiracmeasures?Doestheconcentrationphenomenonoccuronlyaccordingtocertain“quantized”properties?Howdoesthepresenceofboundaryconditionsinfluencetheanswertotheabovequestions? 6MeanFieldEquationsofLiouville-Type6.1PreliminariesInthisChapterweshowhowtoapplytheanalyticalresultsestablishedinChapter5tothestudyofMeanFieldEquationsoftheLiouville-typeoveraclosedRiemannsurface(M,g).Moreprecisely,foragivenµ>0andh∈L∞(M),weconsiderthemeanfieldequation:
'h(x)ew1−gw=µh(x)ewdσg−|M|inM'M(6.1.1)wdσg=0,Mwherewerecallthatganddσgdenote,respectively,theLaplace–Beltramiopera-torandthevolumeelementrelativetothemetricgonM.Weshallbeinterestedinanalyzing(6.1.1)whenhtakestheform$m2αjγh(p)=dg(p,pj)V(p)∈C(M),γ∈(0,1]andV>0,(6.1.2)j=1where{p1,...,pm}⊂Misasetofdistinctpoints,αj>0foreveryj=1,...,m,anddgdenotesthedistancefunctionon(M,g).Duetothestructureof(6.1.1),welosenogeneralityinassumingthathisconve-nientlynormalizedasfollows:!loghdσg=0.(6.1.3)MWeaimtoobtainexistenceresultsfor(6.1.1),aswellas,compactnessforthesolutionset,accordingtothevalueoftheparameterµ>0.AsadirectconsequenceofTheorem5.7.65,wecanclaimthefollowingaboutthesolutionsof(6.1.1). 2506MeanFieldEquationsofLiouville-TypeProposition6.1.1Lethsatisfy(6.1.2)and(6.1.3)with0,1(M):00,suchthateverysolutionwof(6.1.1)withµ∈satisfies||w||C2,γ(M)≤C,γ∈(0,1)andCdependsonlyonb1,b2,A,andmax{µ:µ∈}.AsaconsequenceofProposition6.1.1,weknowthatforeveryµ∈R+,theLeray–SchauderdegreedµoftheFredholmmap,I+µTh,iswelldefinedatzero.RecallthatThisthecompactoperatordefinedin(2.5.6)whichisassociatedtoproblem(6.1.1).Infact,dµisconstantonthefamilyofnumerableopenintervalsInsuchthat∂In⊂and+=∪+∞IRn.(6.1.5)n=1Itisaninterestingopenproblemtocomputeexplicitlythevalueofdµintermsoftheintegern∈N.Forn=1,itiseasytocheckthatI1=(0,8π)anddµ=1∀µ∈(0,8π)(see[L2]).Ontheotherhand,forn≥2anexplicitformulafordµisavailableonlywhenhnevervanishesoverM,namely(inournotation),whenαj=0∀j=1,...,m.Inthiscase,=8πNandIn=(8π(n−1),8πn).Asalreadymentioned,Chen–Linin[ChL2]succeededinexpressingdµintermsofnandtheEulercharacteristicsχ(M)ofMforeveryµ∈In;see(2.5.10).AnequivalentformulaisnotyetavailableforthecasewhenhvanishesatsomepointsonM,asin(6.1.2).Inthiscase,weexpectthetopologicalroleofMtobereplacedwiththepuncturedmanifoldM{p1,...,pm}.ProgressinthisdirectionhasbeenmadebyChen–Lin–Wangin[ChLW],butonlyforthesecondintervalI2in(6.1.5).Infact,assumingthathsatisfies(6.1.2)withαj≥1foreveryj=1,...,m,thenI2=(8π,16π)andasshownin[ChLW]1,∀µ∈(0,8π)dµ=(6.1.6)χ(M)+m+1,∀µ∈(8π,16π)withmthenumberofdistinctzeroesofhinM.Noticethat,ifhnevervanishesonM,then“formally”wecantakem=0in(6.1.6)andseethatitreducesto(2.5.10).Forlateruse,wementionthatbothProposition6.1.1andthedegreeformulaediscussedaboveholdforanelliptic2×2systemthatplaysacrucialroleinthestudyofelectroweakperiodicvortices,asdiscussedinChapter7.Moreprecisely,weconsiderthesystem:⎧

hew11few21⎪⎪−gw1=µ'hew1dσ−|M|+λ'w2−|M|inM,⎪⎨
MgM
fedσgµhew11λfew21gw2=2'hew1dσ−|M|+2'w2−|M|inM,(6.1.7)⎪⎪⎪⎩Mg2cosθMfedσg''w1,w2∈H1(M):w1dσg=0=w2dσg,MM 6.1Preliminaries251whereµ>0,λ>0,θ∈(0,π),andhandfaretwoweightfunctionsofthetype2describedin(6.1.2).Forλ=0,thesystem(6.1.7)decouplesanditssolvabilityisequivalenttothatof(6.1.1).Infact,thetwoproblemsareverymuchrelatedasweseebythefollowing:Corollary6.1.2Lethsatisfy(6.1.2),(6.1.3),and(6.1.4),andlet!γ(M)forγ∈(0,1],00andacompactset⊂R(withasgivenin2(5.7.25))thereexistsaconstantC>0suchthatevery(w1,w2)solutionof(6.1.7)withµ∈andλ∈[0,λ0]satisfies||w1||C2,γ(M)+||w2||C2,γ(M)≤C,whereCdependsonlyonba21,b2,A(in(6.1.4)),,λ0,andµ0=max{µ:µ∈}.a1Proof.Weonlyneedtotreatthecaseλ>0.Byadirectapplicationofthemaximumprincipletothesecondequationof(6.1.7),weseethat!1µcos2θfew2dσmaxw2−logg≤log+1.(6.1.9)MM|M|a1λTherefore,settingfew21gλ=λ'−∈Cγ(M),γ∈(0,1],few2dσg|M|Mforµ∈andλ∈[0,λ0),weseethat!gλdσg=0and||gλ||L∞(M)≤c0,Ma2,λwithc0dependingonlyon0,andµ0.Therefore,ifw0,λdenotestheuniquea1solutionfortheproblem−gw0,λ=gλinM,'w0,λdσg=0,Mthen||w0,λ||C1(M)≤C0∀λ∈(0,λa20],andC0>0depends(asc0)onlyon,λ0,andµ0.Setw=w1−w0,λ,a1whichsatisfies⎧⎨heˆw1− w=µ'−inM,heˆwdσ|M|Mg⎩'w∈H1(M):wdσg=0,M 2526MeanFieldEquationsofLiouville-Typewithhˆ=hew0,λ.Sincehˆkeepsthesamepropertiesofh,wecanapplyProposition6.1.1towandfind||w||C2,γ(M)≤C1,γ∈(0,1),a2,λwithC1>0dependingonb1,b2,A,0andµ0.a1Asaconsequence,weseethatw1isuniformlyboundedinC1(M)-norm.Substi-tutingthisinformationintoproblem(6.1.7)togetherwith(6.1.9),wearrivethedesiredestimatebyabootstrapargument.Thusasabove,forµ∈R+,theLeray–SchauderdegreeoftheFredholmmapassociatedto(6.1.7)iswelldefinedatzero.Moreprecisely,defineFµ,λ:E×E→E×E:µλFµ,λ(w1,w2)=w1+µTh(w1)+λTf(w2),w2+Th(w1)+Tf(w2)22cos2θwhereThisthecompactoperatordefinedin(2.5.6).ByvirtueofCorollary6.1.2,forµ∈R2andλ∈[0,λ0]thereexistsR>0suchthat,theLeray–SchauderdegreeofthemapFµ,λinBR×BRiswelldefinedatzero.HereBRistheballaroundthe'originofradiusRinthespaceE={w∈H1(M):w=0}.Furthermore,bytheMhomotopyinvarianceofthedegree,weseethatdeg(Fµ,λ,BR×BR,0)=deg(Fµ,λ=0,BR×BR,0)=deg(Id+µTh,BR,0)=dµ.Inotherwords,theLeray–SchauderdegreeoftheFredholmmapassociatedtoFµ,λisindependentofλandcoincideswiththedegreeoftheFredholmmaprelativeto(6.1.1).Inparticular,undertheassumptionsofCorollary(5.7.11),foreveryλ≥0wehave1,forµ∈(0,8π)deg(Fµ,λ,BR×BR,0)=(6.1.10)χ(M)+m+1,forµ∈(8π,16π)where,forµ∈(8π,16π),wealsosupposethathsatisfies(6.1.2)withαj≥1∀j=1...,m.Clearly,theformula(6.1.10)reducesto(2.5.10)forthecaseinwhichhnevervanishesinM(i.e.,whenwehaveαj=0∀j=1,...,min(6.1.2)andconsequently=8πN).Asanimmediateconsequenceof(6.1.10),wefindCorollary6.1.3UndertheassumptionsofCorollary6.1.2,problem(6.1.7)alwaysadmitsasolutionforµ∈(0,8π).Furthermore,if(6.1.2)holdswithαj≥1andµ∈(8π,16π),thenproblem(6.1.7)admitsasolutionprovidedthatMhasapositivegenusgorthatM=S2andm≥2. 6.2Anexistenceresult2536.2AnexistenceresultInthissection,wetakeadvantageoftheconcentration/compactnessresultofTheorem5.7.65togiveavariationalconstructionofthesolutionforproblem(6.1.7),asclaimedinCorollary6.1.3whenMadmitsapositivegenus.Thisconstructionwasavailablebeforethedegreeformulain(6.1.10)hadbeenestablishedin[ChLW].ItisbasedonanapproachusedbyDing–Jost–Li–Wang[DJLW3]tohandletheMeanFieldEquation(6.1.1).Inthesamespirit,wementiontheworkofStruwe–Tarantelloin[ST]intheperiodicsetting,concerningtheproblem:
w− w=µ'e−1in=−a,a×−b,b,ew2222'(6.2.1)wdoublyperiodicin∂,w=0,00andifµ∈(8π,16π){8π(1+αj),j=1,...,m}thenproblem(6.1.7)admitsasolutionforeveryλ≥0andθ∈0,π.2Weobservethatproblem(6.1.7)admitsavariationalformulationintheproduct'spaceE×E,whereE={w∈H1(M):w=0}definesaHilbertspaceequippedMwiththeusualscalarproductandnorm.Forevery(w1,w2)∈E×E,wedefinethefunctional:!!tan2θ1|∇2w1Iµ(w1,w2)=gw1|dσg−µlog−hedσg22MM!
!+|∇w122w2g+w2|dσg+λtanθlog−fedσg.(6.2.2)M2MItiseasytocheckthatIµ∈C1(E×E)andthatanycriticalpoint(w1,w2)∈E×EforIµsatisfies:⎛⎞∂I2!w1µtanθ⎜he⎟(w1,w2)ϕ=⎝∇gw1·∇gϕ−µ'ϕ⎠dσg∂w12Mhew1dσg!
Mw1+∇g+w2·∇gϕdσg=0,M2(6.2.3)!
∂Iµw1(w1,w2)ϕ=2∇g+w2·∇gϕdσg∂w2M!2few2ϕ+λtan2θ'dσg=0,Mfew2dσgMforeveryϕ∈E. 2546MeanFieldEquationsofLiouville-TypeInotherwords,criticalpointsofIµdefine(weak)solutionsforthe2×2systemofequations:⎧⎛⎞⎪⎪µw1⎪⎪⎪⎪−g1w1+w2=tan2θ⎝'hew1−1⎠onM,⎪⎪2cos2θ2hedσg|M|⎨⎛M⎞
w2(6.2.4)⎪⎪⎪⎪g1w1+w2=λtan2θ⎝'few2−1⎠onM,⎪⎪22fedσg|M|⎪⎪⎩''Mw1dσg=0=w2dσg,MMandsotheysatisfy(6.1.7).Therefore,onceweobtainacriticalpointofIµinE×Ewefindasolutionto(6.1.7).Forthispurpose,notethatforagivenw1∈E,thereexistsauniquew2∈E,(dependingonw1)whichsatisfies(weakly)thesecondequationin(6.2.4).AsimpleapplicationoftheImplicitFunctionTheoremalsoshowsthatthedependenceofw2onw1isofclassC1.Moreprecisely,Lemma6.2.5ThereexistsaC1-mapγ:E→E,suchthat∂Iµ∗(w,z)=0inEifandonlyifz=γ(w).(6.2.5)∂w2(RecallthatE∗denotesthedualspaceofE.)Proof:Westartwiththefollowing:Claim1:Foreveryw1∈Ethereexistsauniquew2∈Ethatsatisfies:∂Iµ∗(w1,w2)=0inE.(6.2.6)∂w2ToobtainClaim1,wefixw1∈EandobservethatthefunctionalI0(w):=Iµ(w1,w)iscoercive,weaklylowersemicontinuous,andboundedbelowinE.Thecorrespond-ingminimizerw2∈Esatisfies(6.2.6).WeseethattherearenoothercriticalpointsforI0inE.Infact,letz∈EbeanothercriticalpointforI0inE.Thereforeitsatisfies:∂Iµ∗(w1,z)=0inE.∂w2Setψ=z−w2,andconsiderthefunctiong∈C2(R,R)definedasfollows:g(t)=Iµ(w1,w2+tψ),t∈R.Wehaveg˙(0)=˙g(1)=0,and!g¨(t)=2|∇gψ|2dσgM⎛!w2+tψ!w2+tψ2⎞+λtan2θ⎝feψ2dσg−feψdσg⎠,M'w2+tψM'w2+tψfedσgfedσgMM 6.2Anexistenceresult255∀t∈R.Noticethat,byJensens’sinequality(2.5.17),wehave:!w!w2fe2fe'wψdσg−'wψdσg≥0,∀w,ψ∈E;(6.2.7)MMfedσgMMfedσgandsog¨(t)≥0inR.Consequentlyg˙=0isidenticallyzeroin[0,1],andweconcludethatnecessarilyψ=0.Thusz=w2asclaimed.Therefore,ateveryw1∈Ethatthemapγ:E→Eassociatesauniquew2satis-fying(6.2.6)inawelldefinedmanner.Inotherwords,(6.2.5)holds.Toshowthatγ∈C1(E)weapplytheImplicitFunctiontheorem(see[Nir])tothefunctionF:E×E→E∗,∂IµF(w1,w2)=(w1,w2).∂w2Claim2:F∈C1(E×E,E∗)andforevery(w∂F1,w2)∈E×Ethemap:(w1,w2)∂w2definesanisomorphismfromEontoE∗.From(6.2.3)itisstraightforwardtocheckthatFisFrechetdifferentiable,andfor´every(w1,w2),(ψ1,ψ2)∈E×E,wehave∂F∂F∗F(w1,w2)(ψ1,ψ2)=(w1,w2)ψ1+(w1,w2)ψ2∈E,∂w1∂w2with!∂F(w1,w2)ψ1φ=∇gψ1·∇gφdσg,∂w1M!∂F(w1,w2)ψ2φ=2∇gψ2·∇gφdσg,∂w2M!w2w22fefe+λtanθ'w2φψ2−'w2ψ2dσgdσg,MMfedσgMfedσgforeveryφ∈E.Since∀p≥1,themap:w→ewiscontinuousfromEintoLp,weimmediatelyconcludethatF∈C1(E×E,E∗).Moreover,ifweidentifyinthecanonicalwayEwithitsdualspaceE∗,then∂F(w1,w2)canbeidentifiedwithacontinuouslinear∂w2operatorA∈B(E,E)oftheform:A=2I+K,(6.2.8)whereK(dependingonw2only)isacompactlinearmaponE.ItremainstobeshownthatAdefinesanisomorphismontoE.BytheFredholmalternativethisisensuredassoonaswecheckthatKerA=0.Forthispurpose,noticethatforψ∈KerA,wehave:!∂F20==(w1,w2)ψψ=2|∇gψ|dσg∂w2M⎛⎞!w2!w222⎝fe2fe⎠.+λtanθ'w2ψdσg−'w2ψdσgMMfedσgMMfedσgSoby(6.2.7)weimmediatelyobtainψ=0. 2566MeanFieldEquationsofLiouville-TypeAtthispoint,theconclusionthatγ∈C1(E)easilyfollowsfromClaim1andtheImplicitFunctiontheoremappliedaroundeachpair(w1,γ(w1)).Bythesecondequationin(6.2.3),weseethatγ(w)isindependentofµ.Forw∈E,definetherestrictedfunctional:!!tan2θ1w|∇2Jµ(w)=gw|dσg−µln−hedσg22MM!
!w22γ(w)+|∇g+γ(w)|dσg+λtanθlog−fedσg.M2MClearlyJµ∈C1(E),andinviewof(6.2.5),wehavethatwdefinesacriticalpointforJµinEifandonlyifthepair(w,γ(w))isacriticalpointforIµinE×E.Remark6.2.6Noticethatifµ∈(0,8π),thenbytheMoser–Trudingerinequality(2.4.17),JµiscoerciveandboundedfrombelowinE.Acriticalpointiseasilyob-tainedinthiscasebyminimization.Onthecontrary,forµ>8π,acriticalpointforJµcanonlybeobtainedbyintro-ducingamoresophisticated“min-max”constructionthatreliesuponthetopologicalinformationthatMadmitsapositivegenus.Tothisend,noticefirstthatJµismonotonedecreasingwithrespecttoµ;sinceforw∈E,byJensen’sinequality,wehave!hewdσ−g≥1.MAlsonoticethat!−feγ(w)dσg≥1,∀w∈E.(6.2.9)MDenotebyZ={p1,...,pm}⊂Mthefinitesetofdistinctzeroesofh.LetX:M→RlbetheembeddingmapofMintoRll≥3,andlet1⊂MZbearegularsimpleclosedcurvesuchthatitsimage˜1=X(1)linkswithaclosedcurve2⊂RlX(M).Theexistenceof1and2isensuredbythepropertythatMadmitsapositivegenus.Forw∈E,set'Xewdσgm(w)='M∈Rl,(6.2.10)ewdσgMthecenterofmassofw.DenotebyDµthesetofcontinuousmapsg:B1→Esuchthat,forr=|z|,wehave:(i)limJµ(g(z))=−∞r→1−(ii)themap:m(eiτ)=limm(g(reiτ))r→1−definesacontinuousmapfromS1into˜1withnon-zerodegree. 6.2Anexistenceresult257Claim1.Ifµ>8π,thenDµisnotempty.ToestablishClaim1foranyp∈MZ,weintroducethefunctionuε,pdefinedinasmallneighborhoodofp,intermsoftheisothermalcoordinatesystemy=(y1,y2)centeredatp,bytheexpressionεuε,p(y)=log2,ε+σπ|y|2whereσ=h(p)>0.Denotebywε,p∈Etheuniquesolutionfortheproblem:⎧uε,p⎨Xe1−gwε,p=8π'Xeuε,pdσ−|M|onM,'Mg(6.2.11)⎩wε,pdσg=0,MwhereXdenotesastandardcut-offfunctionsupportedinasmallneighborhoodofpinMwhereuε,pisdefined.Clearly,wε,p∈Edependscontinuouslyonε>0andp∈M.Moreover,bymeansofGreen’srepresentationformula,itisnotdifficulttoshowthatforp∈MZandε→0,wehave:21||∇gwε,p||2=32πlog+O(1),L(M)ε!wε,p1log−hedσg=2log+O(1),Mεwε,pe'wε,p→δp,weaklyinthesenseofmeasure.edσgMSee[NT2]and[DJLW1].Wenoticealsothat!wε,p22γwε,p||∇g+γwε,p||2+λtanθlog−edσg≤C,(6.2.12)2L(M)MforasuitableconstantC>0independentofε>0andp∈M.Indeed,inviewof(6.2.11),wefindthatη=γwε,psatisfies⎧uε,p2feη
⎪⎨gη=4π'Xeuε,p+λtanθ'η−14π+λtan2θinM,Xedσg2Mfedσg|M|2M(6.2.13)⎪⎩'ηdσg=0.MTherefore,wefindaconstantA>0(independentofεandp)suchthatmaxη≤A.(6.2.14)MThisestimate(togetherwith(6.2.9))impliesthatηAfee'η≤a2=C1.fedσg|M|M 2586MeanFieldEquationsofLiouville-TypeSince
ηwε,pλ2fe1g+η=tanθ'η−,22fedσg|M|M
wε,pweimmediatelyderivethat||∇g2+η||L2(M)≤C2forasuitableC2>0,independentofεandp,and(6.2.12)follows.Letusdenotebyp=p(τ)τ∈[0,2π]aregularsimpleparametrizationof1.Inviewof(6.2.14)and(6.2.12),wecantheneasilycheckthat,forµ>8π,thefunctionhreiτ:=w1−r,p(τ)r∈[0,1]andτ∈[0,2π)belongstoDµ.Claim1isthusestablished.Atthispoint,wemayfollow[DJLW3]anddefinecµ=infsupJµ(w)(6.2.15)g∈Dµw∈g(B1)asa“good”candidateforacriticalvalueofJµ.Claim2.Ifµ∈(8π,16π),thencµ>−∞.TheproofofClaim2reliesinanessentialway,uponthefollowingimprovedformofMoser–Trudinger’sinequality(seee.g.,[CL1]and[DM]forgeneralizations).Lemma6.2.7
LetS1andS2betwosubsetsofMsatisfyingdist(S1,S2)≥δ0>0,andletγ10∈0,.Foranyε>0,thereexistsaconstantc=c(ε,δ0,γ0)>0suchthat,2forallu∈Esatisfying''eudσgeudσgS1S2'≥γ0,'≥γ0,(6.2.16)eudσgeudσgMMwehave,!!u12edσg≤cexp|∇gu|dσg.(6.2.17)M32π−εMAsimpleapplicationofHolder’sinequalityshowsthat,forµ<16π,Jµisboundedbelowandcoerciveonthesetoffunctionssatisfying(6.2.16).Noticealsothat,foranyg∈Dµthereexistsafunctionw∈g(B1)withm(w)∈2.So,ifbycontradiction,weassumethat(6.2.15)yieldstothevalue−∞,thenwewouldfindasequencewn∈E,withm(wn)∈2,andJµ(wn)→−∞.(6.2.18)Hence,wnmustviolate(6.2.16).Sotheremustexistapointp0∈M,suchthatforeveryδ>0,wehave'ewndσgUδ(p0)'→1,asn→∞,ewndσgM 6.2Anexistenceresult259withUδ(p0)={p∈M:dg(p,p0)<δ}.Consequently,!wnedσg|m(wn)−X(p0)|≤|X−X(p0)|'wn+o(1)edσgUε(p0)M≤sup|X(p)−X(p0)|+o(1),asn→∞.p∈Uδ(p0)Butthisisclearlyimpossible,sincebychoosingδ>0sufficientlysmall,theestimate1aboveimpliesthat|m(wn)−X(p0)|0,independentofn,suchthatif(6.2.20)holds,thenforeveryw∈gn(B1)satisfying(6.2.20),wehave||∇gw||L2(M)≤R.WearegoingtoshowthatthereexistsaPalais–Smalesequence{wn}⊂EforJµatlevelcµ(see(2.3.2))with||∇gwn||L2(M)≤R.Forthispurpose,wearguebycontradiction,andsupposethereexistsδ>0suchthateveryw∈E:||∇gw||L2(M)≤Rand|Jµ(w)−cµ|<δalsosatisfies||Jµ(w)||E∗≥δ.Bymeansofapseudo-gradientflow(seeTheorem2.3.4inChapter2),foreveryε∈(0,δ),thereexistsε∈(0,ε)andanhomeomorphismη:E→Ewiththefollowingproperties:(i)η(w)=w,provided|Jµ(w)−cµ|≥ε;(ii)Jµ(η(w))≤Jµ(w);(iii)if||∇gw||L2(M)≤RandJµ(w)≤cµ+ε,thenJµ(η(w))≤cµ−ε.Letuschoosegn∈Dµ⊂Dµsothat(6.2.20)holds.Inviewofproperties(i)andn(ii)above,wealsohavethatη◦gn∈Dµ⊂Dµsatisfies(6.2.20).Letwn∈gnB1nbesuchthatJµ(wn)=maxJµ(w).BydefinitionJµ(wn)≥cµ,so(6.2.20)andw∈gnB1(6.2.21)holdforwnandimplythat||∇gwn||L2(M)≤R.Furthermore,by(6.2.20)and(6.2.23),fornlargewehave:maxJµ(w)=Jµ(wn)≤Jµn(wn)≤cµn+µ−µn≤cµw∈gnB1+(C+1)(µ−µn)0andb>0,andthustakeasaperiodiccelldomain:()()2ab11=(x,y)∈R:|x|≤,|y|≤"z=at+ibs,t,s∈−,.(6.3.2)2222Werecall,thatinthiscase,theGreen’sfunction(cf.(2.5.11))isidentifiedbyan-periodicfunctionG=G(z)givenas11G(z)=log+γ(z),(6.3.3)2π|z|withγ=γ(z)explicitlydefinedin(2.5.15).Let!u0∈C0(M)suchthatu1h=e0∈L(M):u0dσg=0.(6.3.4)MNoticeinparticularthateu0in(6.3.4)attainsitsmaximumvalueatapointp0∈Mwhereu0iscontinuous.BytheMoser–Trudingerinequality(2.4.17),thefunctional!12−8πlog−eu0+wdσI(w)=||∇w||2g(6.3.5)2L(M)M'isboundedfrombelowinE={w∈H1(M):w=0}.MThus,investigatingtheexistenceofanextremalfortheMoser–Trudingerinequal-itybecomesequivalenttoobtainingaminimumpointforIinE.Inthisdirectionweprove:Theorem6.3.8LetMbeasgivenin(6.3.1)andletu0satisfy(6.3.4).Forp0∈M:u0(p0)=maxu0,weassumethatu0∈C2(Uδ(p0))andthatM8πu0(p0)+>0.(6.3.6)|M|ThenIattainsitsinfimuminE.Remark6.3.9(a)ByvirtueofarecentresultofChen–Lin–Wang[ChLW],weactu-allyknowthecondition(6.3.6)issharp.Infact,ifwechooseasu0theuniquesolutionfortheproblem:u8πin,0=8πδz=0−'||(6.3.7)u0=0,u0doublyperiodicon∂,(see2.5.5),then(6.3.6)isjustmissed,aswehave:8πu0(p0)+=0,forp0:u0(p0)=maxp0.||MItisshownin[ChLW]thatinthiscasethefunctionalIin(6.3.5)cannotattainitsinfimuminE. 6.3ExtremalsfortheMoser–Trudingerinequalityintheperiodicsetting263Morerecentcontributionstowardstheexistenceornon-existenceofcriticalpointsforthefunctionalIin(6.3.5)andtheircharacterizationasminimizerscanbefoundin[LiW]and[LiW1].(b)Theorem6.3.8remainsvalidforageneralsurfaceM,wherecondition(6.3.6)isreplacedbyanequivalentconditioninvolvingtheGausscurvatureofM;see[DJLW1]and[ChL2].Toclarifytheroleofcondition(6.3.6)observethat:Proposition6.3.10LetMin(6.3.1)andu0satisfy(6.3.4).ThenπinfI≤−8π4πγ(0)+maxu0+log()+1EM|M|(6.3.8)−4e−u0(p0)ε2log1u8π2)asε→0+,0(p)++O(εε|M|withγtheregularpartoftheGreenfunction(givenin(2.5.15)).Proof.Toestablish(6.3.8),wearegoingtoconsiderourfunctionsasdoublyperiodicfunctions(withtheperiodiccelldomainin(6.3.2))extendedbyperiodicityovertheplane.Moreover,withoutlossofgenerality,afteratranslationandscalingofcoordi-nates,wecanassumethatp1in(6.3.2)).Forε>00=0∈and||=1(i.e.,b=aandz∈,letε2uwithσ=emaxMu0.ε(z)=log(ε2+πσ|z|2)2Definewεastheuniquesolutionfortheproblem:
eu0+uε1− wε=8π'eu0+uε−||in,'(6.3.9)wεdoublyperiodicon∂,wε=0Withaslightabuseofnotations,set!!I(w12u0+uεε)=|∇wε|−8πloge.2Clearly:infI≤I(wε).Weshallestablish(6.3.8)byshowingthatEI(wε)≤−8π(4πγ(0)+u0(0)+log(π)+1)4212(6.3.10)−εlog(u0(0)+8π)+O(ε)asε→0.σεTothispurpose,onthesetε={z:εz∈}defineeu0(εx)!!2eu0(εx)+uε(εx)=,andAu0+uερε=εε=e=ρε.(6.3.11)(1+σπ|x|2)2ε 2646MeanFieldEquationsofLiouville-TypeWehave!2!!2(8π)||∇w||2=−w w=G(x,y)ρ(x)ρ(y)dxdyLA2!!132π1=32πlog+logρ(x)ρ(y)dxdyA2|x−y|2!!(8π)+γ((x−y))ρ(x)ρ(y)dxdy,A2and!!w1w(x)8πloghe=−16πlog+8πlogh(x)edx.(6.3.12)Toestimatethesecondtermin(6.3.12),observethatw(x)=ρ22w(x)h(x)e(x)(1+σπ|x|)e!22141=ρ(x)(1+σπ|x|)explogρ(y)dy4A|x−y|!8π×expγ((x−y))ρ(y)dy.AThus,byJensen’sinequality(2.5.6),weget!8πlogh(x)ew(x)!1122=32πlog+8πlogA+8πlogρ(x)(1+σπ|x|)A!!418π×explnρ(y)dyexpγ((x−y))ρ(y)dydxA|x−y|A!!132π1≥32πlog+8πlogA+logρ(x)ρ(y)dxdyA2|x−y|2!!(8π)+γ((x−y))ρ(x)ρ(y)dxdyA2!16π2+ρ(x)log(1+πσ|x|)dx.AConsequently,!!16π1I(w)≤−logρ(x)ρ(y)dxdyA2|x−y|!16π2−ρ(x)log(1+πσ|x|)dxA2!!(8π)−8πlogA−γ((x−y))ρ(x)ρ(y)dxdy.2A2 6.3ExtremalsfortheMoser–Trudingerinequalityintheperiodicsetting265Notethat,thefirsttermintheaboveestimatecanbewrittenas!!!!11logρ(x)ρ(y)dxdy=logρ0(x)ρ0(y)dxdy|x−y||x−y|!!1−2logρ0(x)β(y)dxdy|x−y|!!1+logβ(x)β(y)dxdy,|x−y|witheu0(0)eu0(0)−eu0(x)ρ0(x)=andβ(x)=.(1+σπ|x|2)2(1+σπ|x|2)2'Since,1log(πσ)=log1ρ21+σπ|x|2R2|x−y|0(y)dy,wederive:!!!!11logρ0(x)ρ0(y)dxdy=log(πσ)ρ0(x)−ρ0(x)|x−y|2R2!12−ρ0(x)log(1+σπ|x|)2!12+ρ0(x)log(1+σπ|x|)2R2!!1+logρ0(x)ρ0(y)dxdy;R2R2|x−y|and!!!11logρ0(x)β(y)dxdy=log(πσ)β(y)|x−y|2!12−β(y)log(1+σπ|y|)2!!1−β(y)(logρ0(x)dx)dy.R2|x−y|Therefore,!!1logρ(x)ρ(y)dxdy|x−y|!!!1=log(πσ)ρ0(x)−2β(x)−ρ0(x)2R2!!122−ρ0(x)log(1+σπ|y|)+β(x)log(1+σπ|x|)2!12+ρ0(x)log(1+σπ|x|)+R,2R2 2666MeanFieldEquationsofLiouville-Typewith!!1R=logρ0(x)ρ0(y)dxdyR2R2|x−y|!!1+2β(y)(logρ0(x)dx)dyR2|x−y|!!1+logβ(y)β(x)dxdy.|x−y|Weintroducethefollowingnotation:!!α=ρ0(x)β=β(x)!R2!ρ22α¯=0(x)log(1+σπ|x|)β¯=β(x)log(1+σπ|x|).R2''Since2ρ0(x)=1=2ρ0(x)log(1+σπ|x|2),wemaywriteRR!A≡ρ(x)=1−α−β,and!ρ2(x)log(1+σπ|x|)=1−¯α−β¯;andobtain(8π)2I(w)≤−γ(0)−8πlog(1−α−β)28π−log(πσ)(1−2α−2β)(1−α−β)28π16π+(1−2α¯−2β¯)−(1−¯α−β¯)(1−α−β)2(1−α−β)2!!(8π)16π−(γ((x−y))−γ(0))ρ(x)ρ(y)−R.2(1−α−β)2(1−α−β)2ThatisI(w)≤−8π(4πγ(0)+log(πσ)+1)+,(6.3.13)with2α+β=−8πlog(1−α−β)+8πlogπσ1−α−β8π+((α+β)(α+β−2(α¯+β¯)))(6.3.14)(1−α−β)22!!(8π)16π−(γ((x−y))−γ(0))ρ(x)ρ(y)−R,2(1−α−β)2(1−α−β)2and→0as→0. 6.3ExtremalsfortheMoser–Trudingerinequalityintheperiodicsetting267Inordertofindtheexplicitexpressionforas→0,noticethat!2!12α=ρ0(x)=+o();24R2σπR2|x|!!22σ|x1|22σ|x2|2β=−2∂1u0(0)(1+πσ|x|2)2+∂2u0(0)(1+πσ|x|2)2(6.3.15)+O(2)21=Olog.Foramorepreciseexpressionforβ,see(6.3.19)below.Furthermore,!!!log|x−y|ρ0(x)ρ0(y)≤2αlog(2|x|)ρ0(x);R2R2R2!!!ρ0(x)log|x−y|β(y)≤αlog(1+2|y|)β(y)R2!+βlog(2|x|)ρ0(x);R2!!!log|x−y|β(x)β(y)≤2βlog(1+2|x|)β(x).''AndsinceR2log(2|x|)ρ0(x)=o()=log(1+2|x|)β(x),weobtainthat−R≤o(2),as→0.(1−α−β)2Consequently,from(6.3.14),(6.3.15)andthefactthatα¯=o()=β¯,itfollowsthat2!!(8π)2≤+8π(α+β)−(γ((x−y))−γ(0))ρ(x)ρ(y)+o(),2(6.3.16)as→0.Ontheotherhand,!!!2σx(γ((x−y))−γ(0))ρ221(x)ρ(y)=∂1γ(0)22dx1dx2(1+πσ|x|)!2σx+2∂2γ(0)2dx2221dx2(1+πσ|x|)+O(2),(6.3.17)as→0. 2686MeanFieldEquationsofLiouville-TypePuttingtogether(6.3.15),(6.3.16),and(6.3.17),weconclude,6!σx22221≤−4π(8π∂1γ(0)+∂1u0(0))22(1+πσ|x|)!-(6.3.18)σx2+(8π∂2γ(0)+∂2u22220(0))22+O(),(1+πσ|x|)as→0.Itisnotdifficulttocheckthat!2√σx14ππσ=log+a(1+πσ|x|2)2σπ24!2√(6.3.19)σx24ππσ=log+1/a(1+πσ|x|2)2σπ24with!π11111/aarctgya=log−arctg−dy;8a22a220yandfrom(6.3.18)weconclude√42σπ(8π γ(0)+u2),(6.3.20)≤−0(0))log+O(σas→0.Recallingthat γ(0)=1,from(6.3.13)and(6.3.20),wededuce(6.3.10).|M|Atthispoint,toestablishTheorem6.3.8,weshalluseasaminimizingsequenceforI,thefamilywλconstructedinSection4.4ofChapert4(seeRemark4.4.28).ThusforMin(6.3.1),u0satisfying(6.3.4),andλlarge,weobtainafunctionwλsatisfying:⎧⎪⎨ew0+uλ1− wλ=8π'−+fλinM,eu0+wλ|M|(6.3.21)⎪⎩Mwλ∈E,'forasuitablefunctionfλ∈C(M)withfλ=0andsuchthat,asλ→∞:M||fλ||L1(M)=O(bλ),||fλlog|fλ|||L1(M)=O(bλlogλ)(6.3.22)with'1e2(u0+wλ)bλ='M→0andliminfbλlogλ=0.(6.3.23)λ(eu0+wλ)2λ→+∞M 6.3ExtremalsfortheMoser–Trudingerinequalityintheperiodicsetting269InadditionI(wλ)→infI,asλ→+∞,(6.3.24)EandIattainsitsinfimumonEifandonlyifwλisboundedinEuniformlyinλ(seeLemma4.4.27).Noticeinparticularthatifw2,λistheuniquesolutionfortheproblem− w2,λ=fλinMw2,λ∈Ethen||∇w2,λ||L2(M)+max|w2,λ|=O(bλlogλ).MTherefore,alongasequence,(whoseexistenceisensuredbythesecondconditionin(6.3.23))λn→+∞suchthatbλlogλn→0(6.3.25)nweseethat||∇w2,λn||L2(M)+max|w2,λn|→0asn→+∞.(6.3.26)MConsequently,thenewsequencewn=w2,λ−wλ(6.3.27)nnsatisfies
'hnewn1− wn=8πhnewn−|M|M(6.3.28)wn∈Ewherehn=eu0+w2,λn→h=eu0uniformlyinM.(6.3.29)Furthermore,⎧'⎪⎪1||∇wn||2−8πlog−hnewn→infI,⎨2L2(M)MEIattainsitsinfimumonEifandonlyifthesequencewnis(6.3.30)⎪⎪⎩uniformlyboundedinE.Weanalyzewhathappensinthecase||∇w2→+∞,asn→+∞.(6.3.31)n||2L(M) 2706MeanFieldEquationsofLiouville-TypeInthisdirectionweprove:Lemma6.3.11If(6.3.31)holds,thenwnadmitsauniqueblow-uppointp0∈Msuchthath(p)=eu0(p)>0,andhnewn'→δp,weaklyinthesenseofmeasureinM(6.3.32)hnewn0M!hwnmaxwn−logne→−∞,∀D⊂⊂M{p0}(6.3.33)DMProof.Byvirtueof(6.3.28),thecondition(6.3.31)impliesthat!maxhwnwn−logne→+∞asn→∞,(6.3.34)MMandsownmustadmitablow-uppointinM.Weseekacharacterizationofsuchablow-uppointinordertoensurethatitoccursawayfromthezerosetofh.Tothispurpose,recallthatthefunctionalIiscoerciveoverthefunctionsthatsatisfyconditions(6.2.16)inLemma6.2.7.Thus,theremustexistapointp0∈Msuchthat'ewnU'δ(p0)→1,asn→∞(6.3.35)ewnMforanysmallδ>0.Thusp0mustcoincidewithablow-uppointforwn,(possiblyalongasubsequence).Weclaimthath(p0)>0.Tothispurpose,foranysmallδ>0,weuse(6.3.35)todeducethat!!!hwnwnwn−ne≤maxhn+o(1)−e=maxh+o(1)−e,MUδ(p0)MUδ(p0)Mfornsufficientlylarge.Therefore,infI=I(wn)+o(1)E!12wn12=2||∇wn||L2(M)−8πlog−hne+o(1)≥2||∇wn||L2(M)M!−8πlog−ewn−8πlogmaxh+o(1)+o(1),asn→∞.MUδ(p0)Thus,bymeansoftheMoser–Trudingerinequality(2.4.17),wefindaconstantC>0,suchthatlogmaxh≥−C,Uδ(p0)foreveryδ>0.Hence,bylettingδ→0andusingthecontinuityofh,weconcludethath(p0)>0asclaimed. 6.3ExtremalsfortheMoser–Trudingerinequalityintheperiodicsetting271ThisinformationallowsustoapplyProposition5.7.64andarriveatthedesiredconclusion.AsaconsequenceofLemma6.3.11,weseethatifxn∈M:wn(xn)=maxwn.(6.3.36)MThen!xwnn→p0andρn:=wn(xn)−loghne→+∞,asn→+∞(6.3.37)M(possiblyalongasubsequence).Set−ρnsn=e2→0,asn→+∞.Weconsiderthedoublyperiodicfunctionswnandhnextended(byperiodicity)overR2anddefine:2π|snz|2ξn(z)=wn(xn+snz)−wn(xn)−,forz∈n:={x:snx∈}.(6.3.38)|M|Henceξnsatisfies:⎧ξnin⎪⎪− ξn=Unen,⎪⎨ξn(0)=maxξn=0,!n⎪⎪⎪⎩Uξnne=8π,nwith2π|snz|2Un(z)=8πhn(xn+snz)e|M|→8πeu0(p0)>0,(6.3.39)uniformlyinC0(R2).locWecanapplyLemma5.4.21(witha=0andµ=8πeu0(p0)),togetherwithRemark5.4.22,toconcludethatalongasubsequence021ξn→ξuniformlyinC(R)withξ(z)=log(6.3.40)loc(1+πσ|z|2)2andwithσ=eu0(p0).Ontheotherhand,wemayalsowriteξn=un(snz)+2logsn,where!w2π|z|2nun(x)=wn(xn+z)−loghne−,z∈M|M| 2726MeanFieldEquationsofLiouville-Typesatisfies−uuninn=Vneun→8πδVnez=0,weaklyinthesenseofmeasureinand2π|z|22π|z|2Vn(z)=8πhn(xn+z)e|M|→8πeuo(p0+z)+|M|:=V(z),uniformlyin.NoticeinparticularthatV(0)=8πeu0(p0)>0.From(6.3.32)andwell-knownellipticestimates,weseethatwn(z)→4G(z−p0),uniformlyinC0(Mp0),withGtheGreen’sfunction.Thusforun,wecanalsoloccheckmaxun−minun≤C,∂∂forasuitableconstantC>0.Therefore,byvirtueofRemark5.6.53(b),weareinapositiontoapplyLemma5.6.52(withα=0)forξnandobtain:Lemma6.3.12Letξnbegivenasin(6.3.38).Foreveryε>0,thereexistconstantsRε>0,andCε>0,andnε∈N:1ξn(z)≤(4−ε)log+Cε,|z|≥Rε(6.3.41)|z|forn≥nε.Bytheasymptoticdecaypropertiesin(6.3.41),weobtain:Proposition6.3.13Letwnin(6.3.27)satisfy(6.3.31)andletp0begivenasinLemma6.3.11.Thenu0(p0)=maxu0andM!12wninfI=lim(||∇wn||2−8πlog−hne)En→∞
2L(M)M(6.3.42)=−8π4πγ(0)+u2)+1.0(p0)+log(π|M|Proof.Observethat'hnewnwn||∇wn||2=8πM'.L2(M)hnewnMThus,'eu0+wnwn||∇wn||2=8πM'L2(M)eu0+wnM!'(wn−loghnewn)=8πwn(xn)+8πhneM(wn−wn(xn))+o(1)!MUξn=8πwn(xn)+neξn+o(1).n 6.3ExtremalsfortheMoser–Trudingerinequalityintheperiodicsetting273By(6.3.39)andthedecayestimate(6.3.41),wecanusethedominatedconvergencetheoremtopasstothelimitintotheintegralaboveandobtain:!!22Uξnlog(1+πσ|z|)u0(p0)neξn→−8πσ,withσ=e.(6.3.43)22nR2(1+πσ|z|)'log(1+πσ|z|2)Usingtheidentity:σR2(1+πσ|z|2)2=1,weconcludethat2=8πw||∇wn||2n(xn)−16π+o(1),asn→∞.L(M)Consequentlyasn→∞,!!12wnwn||∇wn||2−8πlog−hne=4πwn(xn)−2loghne2L(M)MM−8π+8πlog|M|+o(1).WeuseGreen’srepresentationformulaforwntofind!wn!1'hne1ξn(y)wn(xn)=G(xn,y)=G(xn,xn+sny)Un(y)e2Mhnewn2πnM!!11ξn(y)ξn(y)=logUn(y)e+γ(sny)Un(y)e2πn|sny|!!111ξn(y)ξn(y)=4log+logUn(y)e+γ(sny)Un(y)e.sn2πn|y|nAsbefore,by(6.3.41)wecanjustifythepassagetothelimitintotheintegralsaboveandconclude,!!wn1σlog|y|wn(xn)−2loghne=4log−wn(xn)=4MsnR2(1+σπ|y|2)2(6.3.44)−8πγ(0)+o(1),asn→∞,(σ=eu0(p0)).'σlog|y|Inviewoftheidentity2(1+σπ|y|2)2=−log(πσ),weconcludethat

πI(wn)=−8π4πγ(0)+u0(p0)+log+1+o(1),|M|forlargen.Hence,bylettingn→+∞wefind:

πinfI=−8π4πγ(0)+u0(p0)+log+1.(6.3.45)E|M|Comparing(6.3.8)with(6.3.45),wededucethatnecessarilyu0(p0)=maxu0,(6.3.46)Mandtheproofiscompleted. 2746MeanFieldEquationsofLiouville-TypeProofofTheorem6.3.8.AccordinglytoProposition6.3.13andthepropertiesin(6.3.30),ifIdoesnotattainitsinfimuminE,thennecessarily

πinfI=−8π4πγ(0)+maxu0+log+1.(6.3.47)EM|M|Ontheotherhand,byvirtueofProposition6.3.10,when(6.3.6)holdswecanensurethat

πinfI<−8π4πγ(0)+maxu0+log+1,EM|M|whenceImustattainitsinfimuminEasclaimed.6.4TheproofofTheorem4.4.29Theanalysisoftheprevioussection(withu0in(4.1.3)andN=2),enablesustocompletetheproofofTheorem4.4.29.RecallthatbyvirtueofLemma4.4.27,wehavetoanalyzeonlythecasewherethefunctionalIµ=8π(=I)doesnotattainitsinfimuminE,orequivalentlywhenthefamilyw−)in(4.4.26),(4.4.27),andλ(=wλ(4.4.28)satisfies||wλ||E→+∞,asλ→+∞.(6.4.1)AsintheproofofTheorem6.3.8,inthissituationweareabletofindasequenceλn→+∞,suchthatwn=wλ−w2,λ(wherew2,λisdefinedin(4.4.38))satisfies:nn(6.3.32),(6.3.33),and(6.3.46).Recalling(4.4.22)and(4.4.19)andthatinTheorem4.4.29wetake:v2,λ=wλ+dλ.Thenwecancheckthat,foreverydoublyperiodiccontinuoustestfunctionϕ,thereholds!!λneu0+v2,λn(1−eu0+v2,λn)ϕ=λndλeu0+wλn(1−dλeu0+wλn)ϕnn!eu0+wλn8π!e2(u0+wλn)=8π'u+wλϕ−'2ϕ+o(1)e0nλneu0+wλn!wnhne=8π'ϕ+o(1)→8πϕ(p0),asn→∞hnewnandu0(p0)=maxu0;andsoclaim(4.4.45)isestablished.MItremainstoverifytheuniformlimit,maxeu0+v2,λ→0,asλ→+∞.(6.4.2)Noticethataccordingto(4.4.22),thelimitin(6.4.2)isequivalentto'u0+wλ−logeu0+wλ−logλmaxe→0,asλ→+∞.(6.4.3) 6.4TheproofofTheorem4.4.29275Tothisend,noticethat(6.4.3)certainlyholdsalongasequenceλnchosentosatisfy(6.3.25).Indeedforsuchsequenceλn,wecanusetheanalysisoftheprevioussectionforwn=wλ−w2,λtofindnn'e2(u0+wλn)'h2e2wn!'!22n222(wn−loghnewn)22ξnsn'=sn'=snhne=Uneeu0+wλn2hnewn2n!22112→(8π)σ=(8π)σ,asn→∞,24R2(1+σπ|z|)3'−1(maxwwnwherewerecallthatσ=emax¯u0,sn=e2¯n−loghne),ξnandUnarede-finedin(6.3.38)and(6.3.39)respectively,andthelimitofthelatterintegralisjustifiedby(6.3.40)and(6.3.41).Consequently,recallingthat!e2(u0+wn)'2=o(λn),(see(4.4.19))e(u0+wn)wededuce1→0,asn→∞;thatissn2λn'max¯wn−loghnewn)−logλne→0,asn→∞.(6.4.4)Sinceeu0∈L∞(M),(6.4.3)followsforλ=λn.Thus,toestablish(6.4.3)itsufficestoshowthatinfactalonganysequenceλn→+∞,property(6.3.25)holds.Inotherwords,thefollowingstrongerversionofLemma4.4.27holds:Proposition6.4.14limbλlogλ=0,(6.4.5)λ→∞where'1e2(u0+wλ)bλ='2,λeu0+wλandwλ,isdefinedby(4.4.26),(4.4.27),and(4.4.28).Proof.Tosimplifynotation,weperformatranslationofthecoordinatessothatp0=0∈andu0(0)=maxu0.Werecallthatu0isdefinedby(4.1.3)withN=2;Mthereforeu0attainsitsmaximumvalueatapointawayfromthevortexpoints.We8π=0.Therefore,forε>0andwthushave:−u0(0)+εin(6.3.9),andbythe||argumentsofProposition6.3.10,wefind:!!1|∇w2u0+wεI(wε)=ε|−8πlog−e2(6.4.6)π2≤−8π4πγ(0)+u0(0)+log()+1+O(ε)asε→0.|| 2766MeanFieldEquationsofLiouville-TypeToobtain(6.4.5),wearguebycontradictionandassumethatthereexistasequenceλn→+∞andaconstantβ>0suchthatforwn=wλ,thereholdsn'logλne2(u0+wn)β≥,fornlarge.(6.4.7)'22λneu0+wn(8π)Clearly(6.4.7)impliesthatthesequencewnisunboundedinE.Therefore,thefunc-tionalIµ=8π=Iin(6.3.5)cannotattainitsinfimuminE,andso,

πinfI=−8π4πγ(0)+u0(0)+log+1.(6.4.8)E||Nowrecallthatwλisdefinedbytheextremalproperty:fλ(wλ)=inffλ,wherefλAλisdefinedin(4.4.12)andAλisgivenin(4.4.5).Wewishtouse(6.4.6)togetherwith(6.4.8)toevaluatethefunctionalfλ=λoverwεin(6.3.9),forsuitableε>0.Tothisnpurpose,noticethatforx∈ε={z:εz∈},wehave!!418πwε(εx)=logρε(y)dy+γ(ε(x−y))ρε(y)dyAεεε|x−y|Aεε!11σ=4log+4logdy+8πγ(0)+Rε(x),22εR2|x−y|(1+πσ|y|)whereρεandAεaregivenin(6.3.11),σ=emax¯u0andwheretheremainderRε→0eu0(0)−eu0(y)'21a.e.,asε→0.Sinceβ(x)=(1+πσ|y|2)2satisfiesβ(x)=O(log)(see(6.3.15)),wemayestimateRasfollows:!!1eu0(0)R(x)=4log|x−y|β(y)dy+4logdyR2|x−y|(1+πσ|y|2)2!!8π+(γ((x−y))−γ(0))ρ(y)dy≤4log|x−y|β(y)dyA∩{|y|≤|x|+1}!!log|x−y|βu0(0)1dy+4(y)dy+4elog∩{|x|≤|y|−1}{|x−y|≤1}|x−y|!!+16πmax|γ|≤4log(2|x|+1)βmaxu0log2|y|(y)dy+8edyR2(1+πσ|y|2)2+2πeu0(0)+16πmax|γ|≤O(2log1)log(2|x|+1)+C,as→0,(6.4.9)withC>0asuitableconstant.'
Thus,inviewoftheidentity:4log1σ=2logπσ,we|x−y|(1+πσ|y|2)21+σπ|x|2have1πσw(x)=log+2log+8πγ(0)+R(x),as→0.(6.4.10)41+σπ|x|2From(6.4.9)itfollowsinparticularthat11w(x)≤log+log+C,(6.4.11)4(1+|x|2)2−αforfixedα∈(0,2),asuitableconstantC>0and>0sufficientlysmall. 6.4TheproofofTheorem4.4.29277Hence,!!e2(u0(x)+w(x))dx=2e2(u0(x)+w(x))dx!2(u(x)+log1+2log(πσ)+8πγ(0)+R20(x))=e41+σπ|x|2!1σ2=e4logπσ+16πγ(0)dx+o(1),6(1+σπ|x|2)4as→0.Herewehaveusedestimate(6.4.11)tojustifythepassagetothelimitintotheintegralsign.Analogously,!!eu0(x)+w(x)dx=1e2logπσ+8πγ(0)σdx+o(1),2(1+σπ|x|2)2as→0.Therefore:'⎛'e2u0(0)⎞2(u0(x)+w(x))e1⎝(1+σπ|x|2)4⎠'u(x)+w(x)2=2'u0(0)+o(1)(e0)(e)2(1+σπ|x|2)2
1σ=+o(1),as→0.(6.4.12)23Choosen>0suchthat112σβ1((8π))=(6.4.13)n2λn32logλnandsetw∗=w.nnClearlyn→0asn→+∞,andinviewof(6.4.12)and(6.4.13),wecaneasilycheckthatw∗∈Aλforlargen.From(6.4.6),(6.4.8),andwn=wλ,wehave:nnnO(2)≥I(w∗)−I(w∗n)=fλ(w)−fλ(wn)nnnnn:'32πe2(u0+wn)−8πψ1+1−'λn(eu0+wn)2:'32πe2(u0+wn∗)+8πψ1+1−'λn(eu0+wn∗)2''∗(16π)2e2(u0+wn)e2(u0+wn)≥'−'2λn(eu0+wn)2(eu0+wn∗)2!1dt×:0'∗'232πe2(u0+wn)e2(u0+wn)1+1−t'+(1−t)'λn(eu0+wn∗)2(eu0+wn)2'2(u+w'∗(16π)21e0n)e2(u0+wn)=+o(1)'−'.2λn4(eu0+wn)2(eu0+wn∗)2 2786MeanFieldEquationsofLiouville-TypeThatis,'e2(u0+wn)'e2(u0+wn∗)1'121σ1u+w≤'u+w∗+O(n)=+o.λn(e0n)2λn(e0n)2λnn23logλnTherefore,forlargen,''β(16π)21e2(u0+wn)(8π)2e2(u0+wn)≤aλn=*'2'u+wn)2='u+wn)2logλn2(u0+wn)λn(e0λn(e01+1−32π'eλn(eu0+wn)2⎛⎞'(16π)2e2(u0+wn)⎜11⎟+'⎜⎝*'−⎟⎠λn(eu0+wn)232πe2(u0+wn)4(1+1−'u0+wn)2λn(e)2''(8π)2e2(u0+wn)16πe2(u0+wn)='1+'(1+o(1)λn(eu0+wn)2λn(eu0+wn)2(8π)21σ1≤+o.λnn23logλnThisisclearlyimpossiblebythechoiceofnin(6.4.13),and(6.4.5)isestablished.AsaconsequenceofProposition6.4.14,weseethat(6.4.4)holdsalonganyse-quenceλn→+∞.Therefore(6.4.3)isvalid,andtheproofofTheorem4.4.29iscompleted.6.5FinalremarksandopenproblemsInconcluding,wementionthatitwouldbeextremelyusefultofindextensionsoftheformula(2.5.10)toevaluatethedegreedµoftheFredholmmapassociatedtoproblem(6.1.1),(6.1.2),beyondthecaseµ∈(0,16π){8π}asgivenin(6.1.6)(cf.[ChLW]).Sinceforproblem(6.1.1),(6.1.2)existenceresultsarenotyetavailablewhenµ>16π,alreadyitwouldbeusefultoknowthesurfaceM,forwhichwecanes-tablishthatdµ=0forlargeµ.Aswehaveseen,anyinformationaboutdµcouldbeimmediatelyturnedintoaninformationforthedegreeoftheoperatorassociatedtoproblem(6.1.7),whichentersintotheconstructionofelectroweakvortexconfigurations,asweshallexplaininthenextchapter.Also,asforthepreviouschapter,itwouldberelevanttoextendtheanalysisabovetocoversystems.Needlesstosay,oneofthemaindifficultiesassociatedwithsuchanextensionisthefactthatcrucialtools,suchasthemaximumprinciple,arenolongeravailableforsystems.AlreadyfortheSU(n+1)-Todasystem,whichwehaveseentoariseinanaturalwayinthestudyofnon-abelianChern–Simonsvortices,suchdifficultiesarenoteasytoovercome,evenifweneglectthepresenceoftheDiracmeasures. 6.5Finalremarksandopenproblems279Inthiscontextwehavealreadymentionedthecontributionsin[CSW],[W],[JoW1],[JoW2],[LN],[ChOS],[MN],and[SW1],[SW2].Recently,Jost–Lin–Wangin[JoLW]haveestablishedsuitableextensionsofsomeoftheresultsgivenabovebyadeeperunderstandingofthe“bubbling”phenomenonforsolutionsoftheSU(n+1)-Todasystem. 7SelfdualElectroweakVorticesandStrings7.1IntroductionInSection1.4,wereviewedhowtoattainselfdualityfortheSU(2)×U(1)-electroweaktheoryofGlashow–Salam–Weinberg[La]describedbytheLagrangeandensity(1.4.15).Recallthat,fortheunitarygaugevariablesgivenin(1.4.17)–(1.4.20)andwiththehelpofthevortexansatz(1.4.26)–(1.4.29),wemayformulatethetheoryintermsofacomplexvaluedmassivefieldW,ascalarfieldϕ,andtwo(real)2-vectorfields,P=(Pj)j=1,2andZ=(Zj)j=1,2,whichareassumedtodependonlyonthe(x1,x2)-variables.ThemassivefieldWisweaklycoupledtoPandZthroughthecovariantderivative:DjW=∂jW−ig(Pjsinθ+Zjcosθ)W,j=1,2(7.1.1)wheregistheSU(2)-couplingconstantandθ∈(0,π)istheWeinbergmixingangle2thatrelatestotheU(1)-couplingconstantg∗bymeansoftheidentity:gcosθ=.1(g2+g∗2)2InthiswaytheexpressionforthecorrespondingenergydensityEtakestheform1
g22+P2−2gsinθ|W|2E=|D1W+iD2W|12−ϕ022sinθ
22
21g222g+Z12−ϕ−ϕ0−2gcosθ|W|+ϕZj+εjk∂kϕ22cosθ2cosθ(7.1.2)g2g2
2gϕ2gϕ24+λ−2−ϕ2−0Z0−2ϕ02ϕ012+P128sinθ8cosθ2sinθ2sinθ
g2−∂kεjkZjϕ,2cosθ 2827SelfdualElectroweakVorticesandStringswhereasusual,Z12=∂1Z2−∂2Z1andP12=∂1P2−∂2P1denotethe“curl”ofthevectorfieldZandP,respectively,andεjkdenotesthetotalantisymmetricsymbolwithε12=1.Thus,inthe“critical”coupling,g2λ=,(7.1.3)8cos2θwededucethefollowingselfdualequations:⎧⎪⎪D1W+iD2W=0,⎪⎪⎨g2222P12=(ϕ−ϕ0)+2gcosθ|W|,2sinθ(7.1.4)⎪⎪⎪⎪2cosθ⎩Zj=−εjk∂klogϕ,j=1,2,gwhosesolutionsminimize(7.1.2),wheneverwesatisfyappropriateboundarycondi-tionstoneglectthedivergenceterms.InthisChapter,weshallbeinterestedinestablishingrigorousexistenceresultsfor(7.1.4),intheplanarcaseunderasuitabledecayassumptionatinfinity,andintheperiodiccase,underthe’tHooftperiodicboundarycondition.Wereferto[AO1],[AO2],[AO3],[CM],[V1],[Y5],and[Y7],forotherresultsonelectroweakvortex-likesolutions.Theplanarsolutionsof(7.1.4)overR2havebeenestablishedfirstbySpruck–Yangin[SY1]byashootingmethod,andmorerecentlybyChae–Tarantelloin[ChT1]byaperturbationapproachsimilarinspirittothatintroducedbyChae–Imanuvilov[ChI1]inthestudyofnon-topologicalChern–Simonsvortices(discussedinChapter3).Theconstructionsin[SY1]and[ChT1]yieldtodifferentclassesofsolutionsfor(7.1.4),distinguishedaccordingtotheirasymptoticbehavioratinfinity.Itisnotexcludedthatyetothertypeofsolutionsmayexist.From(7.1.2),weseethatsolutionsof(7.1.4)overR2carryinfiniteenergy,afactthatononehandjustifiestheirabundance,butontheotherhandmakesonewonderabouttheirphysicalinterest.Thus,totreataproblemwithamoredefinitephysicalflavor,weshallfocusonplanarelectroweakstrings,wherewealsotakeintoaccounttheeffectofgravitybycouplingtheelectroweakequationswithEinstein’sequations.Weconsideramodelwherestringsareparalleltothex3-direction,andwherewetakegravitationalmetricstovaryintheclass:
2
2ds2=(dx0)2−(dx3)2−eηdx1−dx2.(7.1.5)Theconformalfactorη=η(x1,x2)iscoupledtotheremainingelectroweakvariablesthroughEinstein’sequations.Correspondingly,inthe“critical”coupling(7.1.3),theselfdualstringequationsbecome: 7.1Introduction283⎧⎪⎪D1W+iD2W=0,⎪⎪g⎪⎪P2η2⎪⎪12=ϕ0e+2gsinθ|W|,⎪⎨2sinθ
g22η2(7.1.6)⎪⎪Z12=2cosθϕ−ϕ0e+2gcosθ|W|,⎪⎪⎪⎪⎪⎪2cosθ⎪⎩Zj=−εjk∂klogϕ,j=1,2,gtobecoupledwithEinstein’sequations,thatunder(7.1.5)reducetogϕ2g
− η=8πG0Pϕ2−ϕ2212+0Z12+4|∇ϕ|,(7.1.7)sinθcosθwithNewton’sgravitationalconstantG>0.Recallingthattheelectroweak-stringenergydensityisgivenbyg2ϕ4g2
2E=0+ϕ2−ϕ2+g2ϕ2|W|2e−η+2e−η|∇ϕ|2,(7.1.8)24cos2θ08sinθwededuce(7.1.7)byobservingthattheGausscurvatureK1e−η ηrelativetoη=−2theRiemannsurface(R2,eηδj,k)satisfiestherelationKη=8πGE+,(7.1.9)wherethecosmologicalconstantisfixedbyEinstein’sequationsasfollows:πGg2ϕ4=0.(7.1.10)sin2θForthecorrespondingselfgravitatingelectroweakstrings,ameaningfulphysicalprop-ertywouldbethattheRiemannsurface(R2,eηδj,k)carriesfinitetotalcurvature;thatis,!Kηηe<+∞.(7.1.11)R2From(7.1.8)and(7.1.9),weseethat(7.1.11)isactuallyequivalenttorequiringthateη∈L1(R2)andthatwesatisfythefinitetotalenergyconditionwithrespecttothevolumeelementof(R2,eηδj,k)asfollows:!Eeη<+∞.(7.1.12)R2Forfurtherdetails,wereferto[Y1]whereactuallytheexistenceofsolutionsforprob-lem(7.1.6),(7.1.7),(7.1.11)islistedasanopenproblem,whichonlyrecentlyhasbeensuccessfullytackledbyChae–Tarantelloin[ChT2].Onthecontrary,moreisknownaboutcosmicstringsrelativetothecouplingofEinstein’sequationswithothergaugefieldstheoriessuchastheMaxwell–Higgstheory,theChern–Simonstheory,etc.(cf.[EGH],[Kb],and[VS]).Inthisrespect,we 2847SelfdualElectroweakVorticesandStringsmentionthecontributionsin[Lint],[V2],[EN],[CG],[CHMcLY],[Y2],[Y4],[Ch4],[ChCh1];andwereferto[Y1]foramoredetaileddiscussionandadditionalreferences.Clearly,ifweneglectthegravitationaleffectbytakingG≡0andη≡0,then(7.1.6)–(7.1.7)reduceto(7.1.4).Infact,itispossibletofollowtheconstructionindi-catedin[ChT1]forelectroweakvorticesinordertoobtainelectroweakstringswiththedesiredfinitecurvatureand,energyproperty(7.1.11)and(7.1.12).Wepresentindetailstheconstructionof[ChT2]inthefollowingsection.7.2PlanarselfgravitatingelectroweakstringsWedevotethissectiontoconstructingafamilyofplanarselfgravitatingelectroweakstrings,correspondingtosolutionsfor(7.1.6)–(7.1.7)andwhichsatisfythefinitetotalcurvatureandenergyconditions,namely,(7.1.11)and(7.1.12),respectively.Moreprecisely,weestablishthefollowing:Theorem7.2.1([ChT2])LetN∈Nsatisfy:sin2θN+1<.(7.2.1)4πGϕ20Foranygivensetofpoints{z1,...,zN}⊂R2(repeatedaccordingtotheirmulti-plicity),thereexistsε1>0suchthatforanyε∈(0,ε1),wefindaselfgravitatingelectroweakstring(Wε,ϕε,Pε,Zε,ηε)solutionof(7.1.6)–(7.1.7),satisfyingthefi-nitecurvatureandenergyconditions(7.1.11)and(7.1.12)withWεvanishingexactlyatthepoints{z1,...,zN}accordingtotheirmultiplicity.ToobtainTheorem7.2.1,weshalluse(2.1.22)and(2.1.23)toreduceourproblemtothesearchforsolutionstotheellipticsystem(2.1.26)(derivedinSection2.1ofChapter2)formulatedintermsofthevariables(u,v,η),where:|W|2=euandev=ϕ2.Moreprecisely,weshallanalyzeinR2,thefollowingellipticsystem:⎧⎪⎪N⎪⎪−u=g2ev+η+4g2eu−4πδz,⎪⎪j⎪⎪j=1⎨2
 v=gev−ϕ2eη+2g2eu,(7.2.2)⎪⎪2cos2θ0⎪⎪2⎪⎪⎪⎪ev−ϕ2ϕ4− η=4πGg2eη0+0+16πGg2eu+v+8πG|∇v|2ev.⎩cos2θ2sinθNoticethat,thegivenpoints{z1,...,zN}(repeatedwithmultiplicity)correspondtothezeroesofthemassivefieldWwhichisgivenasfollows:NuW(z)=exp+iarg(z−zk).2k=1 7.2Planarselfgravitatingelectroweakstrings285Recalling(1.4.18)and(1.4.19)werecovertheremainingvariablesbymeansoftherelationsin(2.1.23)and(2.1.24).Wearegoingtoattackproblem(7.2.2)byaperturbationtechniqueinspiredby[ChI1](seeSection3.4ofChapter3).Therefore,thefirstgoalwillbetointerprettheellipticsystem(7.2.2)asaper-turbationofagivenLiouville-typeoperator.DuetoitsconformalinvariancesuchaLiouvilleoperatoradmitssomedegeneracies.Fortunately,itispossibletoexploitthestructureoftheperturbationterminordertorestoreaninvertibilitypropertyforthe“perturbed”operatorinasuitablefunctionalspace.ThiswillallowustousetheIm-plicitFunctiontheoremandtoobtainasolutionwhosebehaviouratinfinitywecancontrolratherwell.Inthisway,wecanalsocheckthevalidityof(7.1.11)and(7.1.12).Tothispurpose,wetransform(7.2.2)toanequivalentsystemasfollows.Wemul-tiplythesecondequationof(7.2.2)byevandusetheidentityev=ev v+|∇v|2evtodeduce:2
vgv2η+v2u+v2ve=2e−ϕ0e+2ge+|∇v|e.2cosθNextwemultiplytheequationaboveby8πGandaddtheresulttothethirdequationin(7.2.2)tofind:22v2411η4πGgϕ0η+v (η+8πGe)=−4πGgϕ0cos2θ+2e+cos2θe.sinθThus,letting2222411gϕ0λ1=4g,λ2=4πGgϕ0cos2θ+2,λ3=2cos2θ,λ4=8πG,(7.2.3)sinθwearriveatthefollowingequivalentformulationof(7.2.2):Nλ1v+ηuu=−e−λ1e+4πδ(z−zk)(7.2.4)4k=1 (η+λvηη+v4e)=−λ2e+λ3λ4e(7.2.5)λ3v+ηηλ1u2 v=e−λ3e+e,inR.(7.2.6)ϕ220Noticethatthefirstequation(7.2.4)admitsthestuctureofa“singular”Liouvilleequa-tionandthussuggeststhatwetaketheintegrabilityproperty,!eu<+∞,(7.2.7)R2asa“natural”boundarycondition.Since(7.2.7)isscaleinvariantunderthetransfor-mation
x1u(x)−→uε(x)=u+2log,εε 2867SelfdualElectroweakVorticesandStrings∀ε>0,wecanconsidertheε−scaledversionof(7.2.4)–(7.2.6)byalsotransforming:
x1v(x)−→vε(x)=v+2logεε
x1η(x)−→ηε(x)=η+2log.εεIntermsoftheunknowns(uε,vε,ηε),thesystem(7.2.4)–(7.2.6)takestheform:N2λ1v+ηuu=−εe−λ1e+4πδ(z−εzk)(7.2.8)4k=1
2λvη2η+vη+ε4e=−λ2e+ελ3λ4e(7.2.9)ε2λ3v+ηηλ1u2 v=e−λ3e+einR.(7.2.10)ϕ220Thus,wesearchforasolutionof(7.2.8)–(7.2.10)“close”(inasuitablesense)tothoseofthesystem:N0u0u=−λ1e+4πδ(z−εzk)(7.2.11)k=10η0 η=−λ2e(7.2.12)0η0λ1u02 v=−λ3e+einR,(7.2.13)2forwhichwecanexhibitanexplicitsolution.Indeed,asinSection3.4,set$N!zf(z)=(N+1)(z−zk),F(z)=f(ξ)dξ,0k=1andforε>0,let$N!zfε(z)=(N+1)(z−εzk),andFε(z)=fε(ξ)dξ.0k=1By(2.2.3)weseethatthefunctions6-08|fε(z)|208uε,a(z)=log22,ηb(z)=logλ22λ11+|Fε(z)+a|2(1+|z+b|)satisfy(7.2.11)and(7.2.12)respectively,foreveryε>0anda,b∈C.Furthermore,ifweset2λ3κ=,(7.2.14)λ2 7.2Planarselfgravitatingelectroweakstrings287thenwealsosolve(7.2.13)bytaking6-1+|F20ε(z)+a|vε,a,b=log(1+|z+b|2)κ.Reasonably,wemaylookforasolutionof(7.2.4)–(7.2.6)intheform:0(εz)+2logε+ε2σu(z)=uε,a1(εz)(7.2.15)η(z)=η0(εz)+2logε+ε2σb2(εz)(7.2.16)v(z)=v0(εz)+2logε+ε2σε,a,b3(εz)(7.2.17)withσ1,σ2,andσ3suitablefunctionswhichidentifytheerrortermsintheexpansion(7.2.15)–(7.2.17),asε→0.Introducingthenotation:u0(εz)+2logε:=logρI(z),ε,aε,aη0(εz)+2logε:=logρII(z),bε,bv0(εz)+2logε:=logρIII(z),ε,a,bε,a,bweseethatI8ε2N+2|f(z)|2ρε,a(z)=2,a2λ11+ε2N+2F(z)+N+1ε8ε2ρII(z)=,ε,bλ222(1+|εz+b|)22N+2a2ε1+εF(z)+εN+1ρIII(z)=,ε,a,b(1+|εz+b|2)κwhichwemayconsiderfornegativeεaswell.Weprove:Theorem7.2.2LetN∈Nbesuchthat2λ3κ=>N+1.(7.2.18)λ2Foranygivensetofpoints{zj}N∈R2(repeatedaccordingtotheirmultiplicity),j=1thereexistsε1>0,suchthatforeveryε∈(−ε1,ε1),ε=0andproblem(7.2.4)–(7.2.6)admitsasolution(uε,ηε,vε)ofthefollowingform:ε(z)=logρI(z)+ε2w2∗uε,aε∗1(ε|z|)+εu1,ε(εz),(7.2.19)ε(z)=logρII(z)+ε2w2∗ηε,bε∗2(ε|z|)+εu2,ε(εz),(7.2.20)ε(z)=logρIII(z)+ε2w2∗vε,aε∗,bε∗3(ε|z|)+εu3,ε(εz),(7.2.21) 2887SelfdualElectroweakVorticesandStringswithρI∗(z),ρII∗(z),ρIII∗∗(z)definedasaboveand|aε∗|+|bε∗|→0,asε→0.ε,aεε,bεε,aε,bεFurthermore,thefunctionsw1,w2,andw3areradiallysymmetricandsatisfy:w1(|z|)=C1log|z|+O(1)(7.2.22)w2(|z|)=−C2log|z|+O(1)(7.2.23)w3(|z|)=C3log|z|+O(1)(7.2.24)as|z|→∞.TheexplicitconstantsC1,C2,andC3aregiveninLemma7.3.4below;whileu∗,u∗,andu∗satisfy1,ε2,ε3,ε3|u∗(εz)|j,εj=1sup=o(1),asε→0.(7.2.25)21+(log|z|)+z∈RInparticular,euε∈L1(R2),eηε∈L1(R2),and|∇evε|∈L2(R2).(7.2.26)Remark7.2.3Byourconstruction,thesufficientcondition(7.2.18)isclearlyalsonecessarytoensurethevalidityofthelastboundaryconditionin(7.2.26).Noticethatiftheparametersλj,j=1,···,4,areassignedby(7.2.3),then(7.2.18)readsassin2θ>N+1,4πGϕ20andprovidesasufficientconditionfortheexistenceofselfgravitatingelectroweakstrings,asstatedinTheorem7.2.1.Thisconditionisanalogoustothenecessaryandsufficientconditionobtainedin[Y2]fortheexistenceofabelianHiggsstringsintheEinstein–Maxwell–Higgssystem.Itimposesarestrictionbetweenthetotalstringnum-berNandthegravitationalconstantG,whichshouldbeconsideredasasmallpara-meter.Noticealsothatϕ0in(7.2.1)playstheroleofasymmetry-breakingparameteranalogouslytotheabelianHiggsstringsmodel.ClearlyTheorem7.2.1isastraightforwardconsequenceofTheorem7.2.2,andsoweshalldevotethenextsectiontotheproofofthelatter.7.3TheproofofTheorem7.2.2Following[ChI1],wederiveourresultbymakinganappropriateuseoftheImplicitFunctiontheorem(seee.g.,[Nir])overtheHilbertspaces:(!)X222+α2α=u∈Lloc(R)|(1+|x|)|u(x)|dx<∞R2(;;)2,222;u(x);2Yα=u∈Wloc(R)|#u#Xα+;1+α;22<∞1+|x|2L(R) 7.3TheproofofTheorem7.2.2289α∈(0,1),alreadyintroducedin(3.4.11)withtherelativenorms.Sincewearegoingtosearchforsolutions(u,η,v)intheform(7.2.15)–(7.2.17),thenbydirectinspectionweseethatthefunctionsσj,j=1,2,3mustsatisfy: σ1=−λ1gII(z)gIII(z)eε2(σ2+σ3)−λ1gI(z)(eε2σ1−1)(7.3.1)4bε,a,bε2ε,a
 σIIIε2σ3λ2IIε2σ22=−λ4gε,a,b(z)e−2gb(z)e−1εIIIIIε2(σ2+σ3)+λ3λ4gb(z)gε,a,b(z)e(7.3.2)
 σ3=λ3gII(z)gIII(z)eε2(σ2+σ3)−λ3gII(z)eε2σ2−12bε,a,bε2bϕ0
+λ1gI(z)eε2σ1−1,(7.3.3)2ε2ε,awherewehavesetIuε,aIIη0IIIv0gε,a(z)=e,gb(z)=eb,gε,a,b(z)=eε,a,b.Todeterminethetriplet(σ1,σ2,σ3)wearegoingtoconsiderthefreeparametersa,b∈Cintroducedaboveaspartofourunknowns.Weconcentratearoundthevaluesa=0andb=0,anddefinetheradialfunctions:I8(N+1)2r2NII8ρ1=limgε,0=2,ρ2=g0=2,ε→0λ11+r2N+2λ21+r2and1+r2N+2ρgIII=.3=limε,02κε→0(1+r)Thus,bytakinga=b=0in(7.3.1),(7.3.2),and(7.3.3)andbylettingε→0,(formally)weobtainthelinearsystem:λ1 w1+λ1ρ1w1=−ρ2ρ3(7.3.4)4 w2+λ2ρ2w2=−λ4 ρ3+λ3λ4ρ2ρ3(7.3.5)1λ3 w3=λ1ρ1w1−λ3ρ2w2+ρ2ρ3.(7.3.6)2ϕ20Consequently,if(w1,w2,w3)isasolutionof(7.3.4),(7.3.5),(7.3.6)then,underthedecomposition:σj(z)=wj(z)+uj(z),j=1,2,3,(7.3.7) 2907SelfdualElectroweakVorticesandStringswereducetosolvefor(u1,u2,u3)thefollowingimplicitproblem:P1(u1,u2,u3,a,b,ε)=u1+λ1gII(z)gIII(z)eε2(u2+u3+w2+w3)4bε,a,b+λ1gI(z)(eε2(u1+w1)−1)+ w2ε,a1=0,ε
IIIε2(u3+w3)P2(u1,u2,u3,a,b,ε)=u2+λ4gε,a,b(z)e+λ2gII(z)(eε2(u2+w2)−1)ε2bIIIIIε2(u2+u3+w2+w3)−λ3λ4gb(z)gε,a,b(z)e+ w2=0,andP3(u1,u2,u3,a,b,ε)=u3−λ3gII(z)gIII(z)eε2(u2+u3+w2+w3)2bε,a,bϕ0+λ3gII(z)(eε2(u2+w2)−1)ε2b−λ1gI(z)(eε2(u1+w1)−1)+ w2ε,a3=0.2εConcerningthelinearsystem(7.3.4)–(7.3.6),wehave:Lemma7.3.4Forκ>N,thereexistsaradialsolution(w1,w2,w3)of(7.3.4)–(7.3.6)inYα3satisfyingC1w1(r)=C1logr+O(1),andw˙1(r)=+O(1)(7.3.8)rC2w2(r)=−C2logr+O(1),andw˙2(r)=−+O(1)(7.3.9)rC3w3(r)=C3logr+O(1),andw˙3(r)=+O(1)(7.3.10)rasr→∞,where:λ1κ(κ−1)···(κ−N)−(N+1)!C1=,andsoC1>0forκ>N+1;λ2(1+κ)κ···(κ−N)4(λ2+λ3)λ4κ2(κ−1)···(κ−N)+(κ−2N−2)(N+1)!C2=,λ2(2+κ)(1+κ)···(κ−N)andsoC2>0forκ>N+1;C1λ34µC3=−−C2+;2λ2(κ+1)λ2λλ2λ4λrespectively,withµ=3−3−1andκasdefinedin(7.2.14).ϕ2λ280BeforegoingintotheproofofLemma7.3.4,werecallthefollowingpropertiesrelativetotheoperatorsdefinedbytheright-handsideof(7.3.4)and(7.3.5),asestab-lishedinProposition3.4.19. 7.3TheproofofTheorem7.2.2291Proposition7.3.5Forα∈(0,1)andj=1,2,setLj=+λjρj:Yα→Xα.WehaveKerLj=Spanϕj,+,ϕj,−,ϕj,0,(7.3.11)whererN+1cos(N+1)θrN+1sin(N+1)θϕ1,+=,ϕ1,−=,1+r2N+21+r2N+2rcosθrsinθϕ2,+=,ϕ2,−=,1+r21+r21−r2(N+1)1−r2ϕ1,0=,ϕ2,0=.1+r2(N+1)1+r2Moreover,(!)ImLj=f∈Xα|fϕj,±=0.(7.3.12)R2ProofofLemma7.3.4.Takingintoaccount(3.4.26),weknowthataradialsolutioninYroftheequationα w(r)+λ1ρ1w(r)=f(r),(7.3.13)isgivenbytheformula:!2rw(r)=ϕ1,0(r)logr+ϕ1,0(t)f(t)tdt!(N+1)(1+r2(N+1))0r(7.3.14)2−ϕ1,0(r)ϕ1,0(t)logt+f(t)tdt.0(N+1)(1+t2(N+1))'+∞Furthermore,settingcf=ϕ1,0(t)f(t)tdt,thenfromCorollary3.4.21,wealso0knowthatw(r)=−cflogr+O(1),cfw(˙r)=−+O(1),rasr→+∞.Toobtainwλ11,weuseformula(7.3.14)withf(r)=−ρ2(r)ρ3(r).So4λ1Awecancheck(7.3.8)withC1=1where4!∞8!∞(1−r2N+2)rA1=A1(∞)=ϕ1,0(r)rρ2(r)ρ3(r)dr=dr0λ20(1+r2)2+κ!4∞1−tN+141(N+1)!=dt=−λ20(1+t)2+κλ21+κ(1+κ)κ···(κ−N)4κ(κ−1)···(κ−N)−(N+1)!=.λ2(1+κ)κ···(κ−N)So,A1>0forκ>N+1and(7.3.8)isproved. 2927SelfdualElectroweakVorticesandStringsToobtainw2,weusetheanalogousformofformula(7.3.14)fortheoperatorL2,howevernowwetakeN=0andϕ2,0inplaceofϕ1,0.Exactlyasabove,wereducetoevaluate!∞C2=ϕ2,0(r)f(r)rdr,(7.3.15)0withf(r)=λ3λ4ρ2ρ3−λ4 ρ3.Sinceϕ2,0∈KerL2,integrationbypartsleadstotheidentity,!∞!∞!∞ϕ2,0 ρ3rdr= ϕ2,0ρ3rdr=−λ2ϕ2,0ρ2ρ3rdr.(7.3.16)000Consequently,!∞A2=(λ2+λ3)λ4ϕ2,0ρ2ρ3rdr0!∞(1−r2)(1+r2N+2)8(λ2+λ3)λ4=rdrλ20(1+r2)3+κ!4(λ2+λ3)λ4∞(1−t)1+tN+1=dtλ20(1+t)3+κ!6-4(λ2+λ3)λ4∞1ttN+1tN+2=−+−dtλ20(1+t)3+κ(1+t)3+κ(1+t)3+κ(1+t)3+κ4(λ2+λ3)λ411(N+1)!=−+λ22+κ(2+κ)(1+κ)(2+κ)(1+κ)···(1+κ−N)(N+2)!−(2+κ)(1+κ)···(κ−N)4(λ2+λ3)λ4=[(κ+1)κ···(κ−N)−κ(κ−1)···(κ−N)λ2(2+κ)(1+κ)···(κ−N)+(κ−N)(N+1)!−(N+2)!]4(λ2+λ3)λ4[κ2(κ−1)···(κ−N)+(κ−2N−2)(N+1)!]=,(7.3.17)λ2(2+κ)(1+κ)···(κ−N)and(7.3.9)isalsoproved.Inordertoobtainw3(withthegivenasymptoticexpansion),weusethedecompo-sitionw1(r)λ3λ3λ4w3(r)=−+w2(r)+ρ3(r)+ϕ(r),(7.3.18)2λ2λ2whereϕisaregularradialfunctionsatisfying:λ3λ2λ4λ1 ϕ=−3−ρ2ρ3.ϕ2λ280 7.3TheproofofTheorem7.2.2293Setλ3λ2λ4λ1µ=−3−.(7.3.19)ϕ2λ280Incidentally,noticethatthechoiceofλj,j=1,···,4,in(7.2.3)givesµ=g21sin4θ(1+cos2θ).Wehave28µ!r(1+r2N+2)r4µ!r21+tN+1rϕ(˙r)=dr=dtλ20(1+r2)κ+2λ20(1+t)κ+2!r2N+14µ14µt=1−+dt.λ2(κ+1)(1+r2)κ+1λ20(1+t)κ+24µSinceκ>N,rϕ(˙r)→,asr→+∞andthus,λ2(κ+1)4µϕ(r)=logr+O(1).(κ+1)λ2Inviewof(7.3.18),wederivethedesiredconclusionforw3,andwecompletetheproof.Remark7.3.6Withthechoiceof(w1,w2,w3)inLemma7.3.4andforthecondition1,κ−N−1}thereexistsεκ>N+1,weseethatfor0<α02suchthattheoperatorP=(P1,P2,P3)definedaboveisacontinuousmappingfromε={(u,a,b,ε)∈Yα3×C2×R:#u#3+|a|+|b|+|ε|<ε0}intoXα3and0YαP(0,0,0,0,0,0)=0.NextweproceedtocomputethelinearizedoperatorofParoundzero.Fromte-diousbutnotdifficultcomputationsweseethat,fora=a1+ia2andb=b1+ib2,wehave∂gε,Ia(z)∂gε,Ia(z)=−4ρ1ϕ1,+,=−4ρ1ϕ1,−,∂a1∂a2(a,ε)=(0,0)(a,ε)=(0,0)∂gII(z)∂gII(z)b=−4ρ2ϕ2,+,b=−4ρ2ϕ2,−,∂b1∂b2b=0b=0∂gIII(z)∂gIII(z)ε,a,bε,a,b=2ρ3ϕ1,+,=2ρ3ϕ1,−,∂a1∂a2(a,b,ε)=(0,0,0)(a,b,ε)=(0,0,0)∂gIII(z)4λ3∂gIII(z)4λ3ε,a,bε,a,b=−ρ3ϕ2,+,=−ρ3ϕ2,−,∂b1λ2∂b2λ2(a,b,ε)=(0,0,0)(a,b,ε)=(0,0,0)∂gII(z)gIII(z)∂gII(z)gIII(z)bε,a,bbε,a,b=2ρ2ρ3ϕ1,+,=2ρ2ρ3ϕ1,−,∂a1∂a2(a,b,ε)=(0,0,0)(a,b,ε)=(0,0,0)IIIII∂gb(z)gε,a,b(z)λ3=−41+ρ2ρ3ϕ2,+,∂b1λ2(a,b,ε)=(0,0,0)IIIII∂gb(z)gε,a,b(z)λ3=−41+ρ2ρ3ϕ2,−.∂b2λ2(a,b,ε)=(0,0,0) 2947SelfdualElectroweakVorticesandStringsThereforesetting(0,0,0,0,0,0)[vP(u1,u2,u3,a,b)1,v2,v3,α,β]=A[v1,v2,v3,α,β],wecancheckthatforA=(A1,A2,A3),α=α1+iα2andβ=β1+iβ2,wehave:1A1[v1,v2,v3,α,β]= v1+λ1ρ1v1+λ1−4ρ1w1+ρ2ρ3(ϕ1,+α1+ϕ1,−α2)2(7.3.20)λ3−λ1+1ρ2ρ3(ϕ2,+β1+ϕ2,−β2),λ2A2[v1,v2,v3,α,β]= v2+λ2ρ2v2−2λ3λ4ρ2ρ3(ϕ1,+α1+ϕ1,−α2)−2λ4ρ3(ϕ1,+α1+ϕ1,−α2)λ4λ3−4ρ3(ϕ2,+β1+ϕ2,−β2)(7.3.21)λ2λ3−4λ2ρ2w2−λ3λ41+ρ2ρ3(ϕ2,+β1+ϕ2,−β2),λ2andλ1ρA3[v1,v2,v3,α,β]= v3+λ3ρ2v2−1v12+2λ2λ31ρ1w1−2ρ2ρ3(ϕ1,+α1+ϕ1,−α2)(7.3.22)φ0
4λ3λ3+1ρ−4λ3ρ2w1−2λ2ρ3ϕ2,+β1+ϕ2,−β2.φ20Itisinterestingtonotethatalthoughweneedtheconditionκ>N+1toensurethattheoperatorPiswelldefinedfromYα3×C2×(−ε0,ε0)intoXα3,itslinearizedoperatorattheoriginA=(A1,A2,A3)(givenin(7.3.20)–(7.3.22))onlyappearstobewelldefinedfromYα3×C2intoXα3undertheweakerassumptionκ>N,whichalsosufficestoensurethefollowingcrucialproperties:Proposition7.3.7Ifκ>N,thentheoperatorA:(Yα)3×(C)2→(Xα)3givenby(7.3.20)–(7.3.22)isonto.Moreover,

1λ3;ϕ1λ3KerA=Span(0,0,1);ϕ1,±,ϕ2,±,−ϕ1,±+ϕ2,±1,0,ϕ2,0,−ϕ1,0+ϕ2,0;2λ22λ2

ϕ1λ31λ31,±,ϕ2,0,−ϕ1,±+ϕ2,0;ϕ1,0,ϕ2,±,−ϕ1,0+ϕ2,±×{(0,0)}.2λ22λ2(7.3.23)Inordertoprovethestatementabove,weestablishthefollowing:Lemma7.3.8Letκ>N.Then!±122πI1:=−4ρ1w1+ρ2ρ3ϕ1,±dx=,(7.3.24)R22λ2(κ+1) 7.3TheproofofTheorem7.2.2295and!±λ32I2:=−λ2ρ2w2+λ3λ4+1ρ2ρ3ϕ2,±dxR2λ2!λ3λ4− (ρ3ϕ2,±)ϕ2,±dxλ2R2πλ4(N+1)!(N+1)=,(7.3.25)(1+κ)κ···(1+κ−N)withw1andw2asgiveninLemma7.3.4.Proof.Weprove(7.3.24)byrecallingtheformula116(N+1)2r4N+2L1=,(1+r2N+2)2(1+r2N+2)4andcomputing!!()2π∞1r2N+2cos2(N+1)θI±=ρ1−4ρ1w1+2ρ32N+222rdrdθ002(1+r)sin(N+1)θ!6-∞32(N+1)2r2N1r2N+2=π−w1+ρ2ρ3rdr0λ1(1+r2N+2)22(1+r2N+2)2!.∞21ρ2ρ3r2N+2=π−L1w1+rdr0λ1(1+r2N+2)22(1+r2N+2)2!.∞2L1w1ρ2ρ3r2N+2=π−+rdr0λ1(1+r2N+2)22(1+r2N+2)2!.∞ρ2ρ3ρ2ρ3r2N+2=π+rdr02(1+r2N+2)22(1+r2N+2)2!∞!∞πρ2ρ34πrdr2π=rdr==,20(1+r2N+2)λ20(1+r2)κ+2λ2(κ+1)wheretheintegrationbypartsperformedaboveisjustifiedbyvirtueoftheasymptoticbehaviorofw1anditsderivativeasr→+∞,asprovidedinLemma7.3.4.Inordertoprove(7.3.25),againweuseintegrationbypartstoobtain:!!±λ32λ3λ4I2=−λ2ρ2w2+λ3λ41+ρ2ρ3ϕ2,±dx−ρ3ϕ2,± ϕ2,±dx!R2λ2λ2R2(7.3.26)λ32=−λ2ρ2w2+λ3λ42+ρ2ρ3ϕ2,±dx,R2λ2whereweused− ϕ2,±=λ2ρ2ϕ2,±.Inviewoftheidentity116r2L2=,(1+r2)2(1+r2)4 2967SelfdualElectroweakVorticesandStringswemaytransformthefirsttermofI±asfollows2!!∞!2πr2(2)2cosθλ2ρ2w2ϕ2,±dx=−λ2ρ2w222sin2θrdrdθR200(1+r)!∞2!∞rπ1=−8πw2rdr=−L2w2rdr0(1+r2)420(1+r2)2!∞πL2w2=−rdr20(1+r2)2!∞π1=−[λ3λ4ρ2ρ3−λ4 ρ3]rdr,20(1+r2)2whereweused(7.3.9)toderivethelastidentity.Substitutingthisresultinto(7.3.26),wefind!∞!∞±πρ2ρ3π ρ3I=−λ3λ4rdr+λ4rdr220(1+r2)220(1+r2)2!λ3∞ρ2ρ3r3+πλ3λ42+dr=J1+J2+J3.λ20(1+r2)2WecanrewriteJ1andJ3asfollows:!∞π2J1=−λ2λ3λ4ρ2ρ3rdr,(7.3.27)160!πλ3∞ρ2ρ3J3=λ2λ3λ42+23rdr.(7.3.28)8λ20Alsowecaneasilycheckthat ρ222=λ2(2r−1)ρ2.Therefore,forκ>Nwecanperformintegrationbypartsandobtain!∞!∞ππJ2=λ2λ4 ρ3ρ2rdr=λ2λ4ρ3 ρ2rdr160160!∞π232=λ2λ4(2r−r)ρ2ρ3dr.(7.3.29)160Consequently,!π∞λ3I±=J3221+J2+J3=λ2λ3λ44+2r−rρ2ρ3dr160λ2!∞π232+λ2λ4(2r−r)ρ2ρ3dr160!∞π232=λ2λ4κ(4+κ)r−rρ2ρ3dr(7.3.30)320!∞π232+λ2λ4(2r−r)ρ2ρ3dr160π2=λ2λ4(κ+2)[(κ+2)K1−K2],32 7.3TheproofofTheorem7.2.2297where!!∞∞r3ρ2ρ2K1=23drandK2=rρ2ρ3dr.00Weevaluate64!∞r3(1+r2N+2)K1=drλ20(1+r2)4+κ26-!∞!∞N+232tt=dt+dtλ20(1+t)4+κ0(1+t)4+κ2321(N+2)!=+,(7.3.31)λ2(3+κ)(2+κ)(3+κ)(2+κ)···(1+κ−N)2and!∞2N+264r(1+r)K2=drλ20(1+r2)4+κ26-!∞!∞N+1321t=dt+dtλ20(1+t)4+κ0(1+t)4+κ2321(N+1)!=+.(7.3.32)λ23+κ(3+κ)(2+κ)···(2+κ−N)2Substituting(7.3.31)and(7.3.32)into(7.3.30),weobtain±1(N+2)!1I=π(κ+2)λ4+−23+κ(3+κ)(1+κ)κ···(1+κ−N)3+κ(N+1)!−(3+κ)(2+κ)···(2+κ−N)π(κ+2)λ4(N+1)![(N+2)(2+κ)−(1+κ−N)]=(3+κ)(2+κ)···(1+κ−N)πλ4(N+1)!(N+1)=.(1+κ)κ···(1+κ−N)TheproofofLemma7.3.8iscompleted.ProofofProposition7.3.7.Givenf=(f1,f2,f3)∈(Xα)3,weneedtoshowthesolvabilityinYα3×C2ofthelinearequation:A[v1,v2,v3,α,β]=f.(7.3.33)Equivalently,1L1v1+λ1−4ρ1w1+ρ2ρ3(ϕ1,+α1+ϕ1,−α2)2(7.3.34)λ3−λ1+1ρ2ρ3(ϕ2,+β1+ϕ2,−β2)=f1,λ2 2987SelfdualElectroweakVorticesandStringsL2v2−2λ3λ4ρ2ρ3(ϕ1,+α1+ϕ1,−α2)−2λ4[ρ3(ϕ1,+α1+ϕ1,−α2)]λ3−4λ2ρ2w2−λ3λ4+1ρ2ρ3(ϕ2,+β1+ϕ2,−β2)λ2(7.3.35)λ4λ3−4[ρ3(ϕ2,+β1+ϕ2,−β2)]=f2,λ26-λ12λ3 v3+λ3ρ2v2−ρ1v1+2λ1ρ1w1−ρ2ρ3(ϕ1,+α1+ϕ1,−α2)2ϕ26-0(7.3.36)4λ3λ3−4λ3ρ2w1−+1ρ2ρ3(ϕ2,+β1+ϕ2,−β2)=f3.ϕ2λ20Bytheorthogonalitypropertyofthesystem{ϕ1,±,ϕ2,±}andby(7.3.24),wecanex-plicitlydetermine!!λ2(κ+1)λ2(κ+1)α1=−f1ϕ1,+,α2=−f1ϕ1,−2πλ1R22πλ1R2in(7.3.34),andthusensurethat(L1v1,ϕ1,±)L2=0.(7.3.37)Similarlyby(7.3.25),wecandetermineβ1,β2in(7.3.35)suchthat,(L2v2,ϕ2,±)L2=0.(7.3.38)Withsuchchoicesforα1,α2andβ1,β2weareinapositiontouseProposition7.3.7,toobtainthev1,v2∈Yα,solution,respectively,to(7.3.34)and(7.3.35).Atthispoint,set6-λ12λ3g=−λ3ρ2v2+ρ1v1−2λ1ρ1w1−ρ2ρ3(ϕ1,+α1+ϕ1,−α2)2ϕ26-0(7.3.39)4λ3λ3+4λ3ρ2w1−+1ρ2ρ3(ϕ2,+β1+ϕ2,−β2)+f3∈Xα,ϕ2λ20andobservethat(7.3.36)issolvableinYαwiththecorrespondingsolutiongivenby!1v3(x)=log(|x−y|)g(y)dy+C(7.3.40)2πR2foraconstantC∈R.SotheoperatorAisonto.Furthermore,KerAcanbedeterminedbylettingf1=f2=f3=0intheaboveargument.Thisleadstoα1=0=α2,1vλ3β1=0=β2andv3=−1+v2+Cwithvj∈KerLj,j=1,2,andany2λ2constantC∈R.Therefore,thedesiredconclusion(7.3.23)followsoncewetakeintoaccountProposition7.3.7. 7.3TheproofofTheorem7.2.2299ProofofTheorem7.2.2.Wedecompose(Yα)3×C2=Uα⊕KerAwithUα=(KerA)⊥,sothat(0,0,0,0,0,0):U3A=P(u1,u2,u3,a,b)α→(Xα)definesanisomorphism.ThestandardImplicitFunctiontheorem(seee.g.,[Nir])ap-pliestotheoperatorP:Uα×(−ε0,ε0)→(Xα)3forsufficientlysmallε0>0,andimpliesthatthereexistε1∈(0,ε0)andacontinuousfunction,ε%→ψ∗,u∗,u∗,a∗,b∗),ε=(u1,ε2,ε3,εεεfrom(−ε1,ε1)intoaneighborhoodoftheorigininUαsuchthat∗,u∗,u∗,a∗,b∗,ε)=0,forallε∈(−εP(u1,ε2,ε3,εεε1,ε1),andu∗=0foreveryj=1,2,3,anda∗=0=b∗.Consequently,j,ε=0ε=0ε=0ε(z)=logρI(z)+ε2w2∗uε,aε∗1(εz)+εu1,ε(εz),ε(z)=logρII(z)+ε2w2∗ηε,bε∗2(εz)+εu2,ε(εz),(7.3.41)ε(z)=logρIII(z)+ε2w2∗vε,aε∗,bε∗3(εz)+εu3,ε(εz),definesasolutionforthesystem(7.2.4)–(7.2.6),∀ε∈(−ε1,ε1),ε=0.Furthermore,fromLemma3.4.18wehave∗(x)|≤C#u∗#++|uY(log|x|+1)≤C#ψε#U(log|x|+1),j=1,2,3,j,εj,εααwith#ψε#U→0,asε→0.αTherefore,|u∗(εx)|j,εsup=o(1)(7.3.42)1+log+|x|R2asε→0.Since(7.2.18)holds,theexplicitformofρI∗(z),ρII∗(z),ρIII∗∗(z),ε,aεε,bεε,aε,bεtogetherwiththeasymptoticbehaviorsofw1,w2,w2,asdescribedinLemma7.3.4and(7.3.42),implythatthesolution(uε,ηε,vε)in(7.3.41)alsosatisfiestheintegralcondition(7.2.26).Theproofiscompleted.FinalRemarks:ByamorecompleteapplicationoftheImplicitFunctiontheorem(e.g.,[Nir]),wecanactuallyclaimtheexistenceofafamilyofsolutionsthatdependuponanumberofparametersbeingequaltothedimensionofKerA.Aminormod-ificationoftheproofpresentedaboveallowsustoincludeanequalityin(7.2.18).Inthiscase,theimageoftheoperatorPismappedintothespace(Xα−δ)3forδ0>00sufficientlysmall.Noticethat,accordingtoLemma7.3.4,thefunctionswj,j=1,2areboundedinthiscase,(i.e.,C1=C2=0)whilew3divergesatinfinitywithloga-rithmicgrowth.Asaconsequence,theresultingstringsolutionnolongeradmitsfiniteenergy. 3007SelfdualElectroweakVorticesandStringsItisaninterestingopenquestiontoknowwhetherornotproblem(7.2.4),(7.2.5),and(7.2.6)admitsasolutionwhen(7.2.18)isviolated,ormoreprecisely,when2λ3N.However,inthiscase,weseethatthefunctionwweakerassumption:3λ2admitsapowergrowthatinfinity,andsoitfailstobelongtoYα.Therefore,amodifiedfunctionalframeworkisrequiredinordertohandlethissituation.Ontheotherhand,theabovediscussionindicatesthat,asfarasselfgravitatingelectroweaksolutionsaresin2θconcerned,theconditionN+1<2seemstobenecessarywhenwewishto4πgϕ0guaranteethefiniteenergyproperties(7.1.11)and(7.1.12).7.4PeriodicelectroweakvorticesInthissectionwefocusourattentiononsolving(7.1.4)underthe’tHooftperiodicboundaryconditions,overtheperiodiccelldomain={z=a1t+sa2s,00andWvanishesexactlyatzj(accordingtoitsmultiplicity)j=1,...,N.Furthermore,itadmitsatotalflux=2πN(see(2.1.39)),where−e=e−(gsinθ)istheelectriccharge.Theorem7.4.9wasfirstestablishedbySpruck–Yangin[SY2],underthemorerestric-4πN−g2ϕ2||tiveassumption:0<1.8πsin2θIntheformstatedabove,Theorem7.4.9isduetoBartolucci–Tarantello(cf.[BT2]).4πN−g2ϕ2||Stillwhen0=1.thequestionofsolvabilityof(7.1.4)and(2.1.38)in8πsin2θstandsasachallengingopenproblem.Asamatteroffact,eachtimewehave:4πN−g2ϕ2||0∈N,(7.4.3)8πsin2θwefaceadditionalanalyticaldifficultiesinthesolvabilityof(7.1.4)and(2.1.38),forreasonsthatwillbecomeclearinthesequel.Indeed,wesuspectthatonlythevaluesin(7.4.3)representaseriousobstractiontothesolvabilityof(7.1.4)and(2.1.38);andinfact,Theorem7.4.9shouldholdwith(7.4.2)replacedbythecondition:4πN−g2ϕ2N0∈/N.(7.4.4)8πsin2θProofofTheorem7.4.9.Oncemore,wetakeadvantageoftheequivalentellipticfor-mulationcorrespondingto(7.1.4)and(2.1.38),asderivedinSection2.1ofChapter2.Moreprecisely,definingnewvariables(u,v)suchthateu=|W|2ev=ϕ2,weneedtosolve:⎧⎪⎪N⎪⎪2eu+g2ev−4πδ−u=4gzin,⎪⎪j⎨j=1g2(7.4.5)⎪⎪− v=(ϕ2−ev)−2g2euin,⎪⎪20⎪⎪2cosθ⎩u,vdoublyperiodicon∂,inordertorecoverthewholevortex(W,ϕ,P,Z)solutionfor(7.1.4)and(2.1.38),bymeansof(2.1.22),(2.1.23),(2.1.24)and(1.4.18)–(1.4.19).Concerning(7.4.5),observethatuponintegrationover,everysolutionpair(u,v)mustsatisfythecontraints:!!!2!g2ϕ22u2v2ugv04ge+ge=4πN,2ge+e=||.2cos2θ2cos2θ 3027SelfdualElectroweakVorticesandStringsConsequentlywederive,!22!2224πN−gϕ||gϕ||−4πNcosθeu=0;ev=0.(7.4.6)4g2sin2θg2sin2θFrom(7.4.6)weimmediatelydeduce(7.4.1)asanecessaryconditionforthesolvabilityof(7.1.4)and(2.1.38),andpart(i)isestablished.Next,weresumethefunctionu0in(4.1.3),thattogetherwithuandv,wearegoingtoconsiderasfunctionsdefinedovertheflat2-torusM=R2
a1Z×a2Z.Inthisway,wedecompose!!u=u0+w1+c:w1=0andc=−u,!M!M(7.4.7)v=w2+d:w2=0andd=−v.MMFrom(7.4.6)itfollowsthat:4πN−g2ϕ2||1ec=0'anded4g2sin2θeu0+w1Mg2ϕ2||−4πNcos2θ1=0'.(7.4.8)g2sin2θew2MTherefore,lettingg2ϕ2||−4πNcos2θ4πN−g2ϕ2||µ=4πN−0=0,(7.4.9)sin2θsin2θthenecessarycondition(7.4.1)thenreadsas0<µ<4πN.(7.4.10)Problem(7.4.5)cannowbestatedintermsofthenewunknowns(w1,w2)equivalentlyasfollows:⎧eu0+w1ew24πN⎪⎪⎪⎪− w1=µ'+(4πN−µ)'−inM⎪⎨eu0+w1ew2|M|MMµeu0+w114πN−µew21− w2='−+'−inM(7.4.11)⎪⎪2Meu0+w1|M|2cos2θMew2|M|⎪⎪!!⎪⎩1w1,w2∈H(M):w1=0=w2MMClearly,problem(7.4.11)isaparticularcaseofthegeneralellipticsystem(6.1.7),whereM=R2
a1Z×a2Z,λ=4πN−µ>0,h=eu0andf=1.ThereforewecansimplyapplyCorollary6.1.3toobtainasolutionfor(7.4.11),providedthatµ∈{(0,8π)∪(8π,16π)}∩(0,4πN).(7.4.12) 7.5Concludingremarks303Recallingthedefinitionofµin(7.4.9)from(7.4.12)weobtainthedesiredexistenceresultasstatedin(ii).Finallywenoticethat,forN=1,2,3,4,ourexistenceresultisrather“sharp”aswehave:4πN−g2ϕ2||Corollary7.4.10IfN=1,2orN=3,4andif0=1,thencondition8πsin2θ(7.4.1)isnecessaryandsufficientfortheexistenceofaselfdualelectroweakperiodicN-vortex.Moreover,theN-points(countedwithmultiplicity)whereWvanishescanbearbitrarilyprescribed.Thus,weconcludeourdiscussionaboutelectroweakvorticeswiththefollowing:Openquestion:DoestheconclusionofCorollary7.4.10alsoholdforN≥5,possiblybyassuming(7.4.4)?7.5ConcludingremarksWeendthismonographbydrawingthereader’sattentiontootherquestionsofinterestinthestudyofgaugefieldvorticeswhichwerenottoucheduponbyouranalysis.Firstly,weobservethatallofthestaticselfdualvortexproblemsconsideredherecouldbemoregenerallyaddressedoncompactsurfaces(cf.[KiKi])otherthantheflat2−torus,onwhichwehaveinfactfocusedsinceitisencounteredmoreofteninphysicalapplications.Actuallyitmakesgoodsensetoanalyzetheselfdualequationsovercompactsur-faceswithaboundary,whereweassignDirichletboundaryconditionsoffixed“nor-mal”state.Secondly,concerningselfdualChern–Simonsvortices,itisimportanttoclarifywhethertheselfdualsolutionsconsideredhereinfactdescribeallfiniteenergyso-lutionsofthestaticChern–Simonsfieldequations:(1.2.33)–(1.2.34);(1.2.56);or(1.3.103)–(1.3.104)(subjectto(1.3.116)–(1.3.117)).ThiswouldimplythatChern–Simonsmixedvortex/antivortexconfigurationsarenotallowed,asindeedistruefortheMaxwell–Higgsmodel.Ifnot,seewhetherthisisactuallythecaseforacertainre-strictedclassofsolutions,suchaslocalenergyminimizersor“symmetric”solutions,asithappensforYang–Millsfields.Evenmoreimportantly,itwouldbehelpfultoaccuratelydescribethenatureofChern–Simonsvorticesawayfromtheselfdualregime,aswasdoneforGinzburg–Landauvortices(cf.[BBH],[DGP]and[PR]).Forsomeattemptinthisdirection,see[HaK],[KS1]and[KS2].Finally,ourmodelsarebynaturebi-dimensional.Totreathigherdimensionalsitu-ations,itisnecessarytoconsiderothermodelswithsimilarcharacteristicsforexample,themuchacclaimedSeiberg–Wittenmodeldiscussedin[DJLPW]. 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Indexabelian–Higgsmodel,129vortices,7,29,77,111,127,131,162,171,abelian–Higgsstring,288303Abrikosov,40,48Chern–Simons–HiggsAdjontrepresentation,14,15,19,266th-ordermodel,1,8,75,169asymmetricvacuum,76concentration-compactnessprinciple,184,241,248Bianchiidentity,2,28condensedmatterphysics,6blow-upconformalfactor,40,48point,181,182,194,243,270conformalfieldtheory,6profile,227conjugaterepresentation,15,33set,181,195connection,17Bogomolnyi,1,5Cooperpairs,4,75Born–Infieldtheory,111cosmicstrings,111,283Bosonicsector,36Coulumbgauge,43Brouwerdegree,52covariantderivative,17,60CP(1)-model,164Cartancriticalmatrix,24,31,33,46,70,164point,56,136subalgebra,20,22–24value,56Cartan–Weylcurrent,3,10,31,32,34basis,19,20curvature,2,17,28,283,284generators,26,33Cauchy–Riemannequations,12,43deRahmcohomology,29charge,3Diracmeasures,44,53,248Chernnumber,28Dirichletboundaryconditions,247,303Chern–Pontryaginclass,29“double-well”potential,3Chern–Simonsenergy,8Einstein’sequations,40,282,283fieldequations,127electriclimit,129,164charge,9,49,50,81,127,171model,12,33,49,129field,3parameter,6,8,9electromagneticinteractions,2,3,19,37theory,1,6,283electromagnetism,19 324IndexElectroweakinstantons,5,28strings,40,41,48,51,282,284internalsymmetry,13,18theory,1,19,35vortices,40,47,50,111,174Jacobiidentity,19ellipticregularity,76,114,177Jensen’sinequality,67,132,140,158,160,energy,4,32193,256,264Eulercharacteristic,68Euler–Lagrangeequations,3,8,10,30,31Killingform,15,21,27exponential,16,35,52lagrangean,3,6,10,18,30Flattorus,64Laplace–Beltramioperator,64,242,249Fourierdecomposition,114Laplacianoperator,52Fredholmmap,66,67,278Leray–Schauderdegree,67,252Liealgebra,13,14,18,21gaugebracket,14,27field,2,13,17,18,36group,1,13,14,17,18group,13Liovilleinvariance,34,43equation,52,60,112,212,285map,18formula,65,112,158,212,220potential,2,17problem,111theory,1,17transformations,3,12,38,43magneticGaussfield,3,171curvature,41,51,243,263,283flux,5,49,50,81,127law,4,7,11,32massscaleparameter,10,31genus,253,256massivefield,284,300Ginzburg–Landaumaximalsolution,92,96,137,144model,1,13,52maximumprinciple,76,132,251theory,4Maxwellvortices,6,13electrodynamics,6gravitationalconstant,283equations,4,28gravity,40Maxwell–Chern-Simons–Higgs(MCSH)Green’smodel,49,129,130function,68,247,263,272theory,9,10representationformula,86,160,196,228,vortices,11,12,129239Maxwell–Higgsequations,79Harnackinequality,107,177,178,200,213limit,163heatflow,145model,1,7,12,29,49,76,129,303Hermitianconjugate,16theory,19,283Higgsfield,2,12,17,18,27,81,129vortices,76Hodgeoperator,28meanfieldequation,64,249,253holomorphic,43,52,53,220metrictensor,2homotopy,29minimalsolution,137,138,144MinkowskiImplicitFunctionTheorem,113,254,285,metric,7299space,2,17,35inf+supestimates,206,237min-max,67,256 Index325momentum,7specialmonopoles,30orthogonalgroup,16Moser–Trudingerinequality,64,140,152,unitarygroup,16155,158,171,177,256,261static,4,5“mountain-pass”,56,103,139,141,145stereographicprojection,62“movingplanes”,55,212stronginteractions,19subharmonicfunction,85Neumannboundaryconditions,247subsolution,92,136–138neutralscalarfield,9,10,12SU(n+1)-vortices,72,165Nielsen–Olesenvortex,5,29superconductivity,1,4,75non-abelian,1,5,19,31,33,130,164superharmonicfunction,94non-relativistic,1,12,34supersolution,134,137non-topological,65,76,111,126,129,130,symmetricvacuum,76145,165,168,248symmetrybreaking,10,31,54,288orderparameter,4temporalgauge,4,7,28,29’tHooft,39,48,51,300Palais–Smale,56,57Thetafunction,69Paulimatrices,25,35time-dependent,9periodicity,48TodaPohozaevtypeidentity,100,179,196,234-functional,172potentialfield,2,36-system,72,248,278principaltopologicaldegree,49bundle,17topologicalsolutions,126embeddingvacuum,130,164topologicalvortices,76,81,128pseudo-gradient,57,145totalenergy,49,81,127,283trace,15,23quantumHalleffect,6triplewellpotential,9radiallysymmetric,100,127unbrokenvacuum,130,164rank,20,129unifiedtheory,19Riemannunitarygauge,37,39metric,60surface,41,61,249,283vortexroot,19,20number,51,76,77,300points,51,75,77,137,173scalarpotential,3Seiberg–Wittenmodel,303W-condensates,39selfdualweakinteractions,19,37equations,5,8,11,33,39Weierstrassfunction,65,158regime,8,13Weinbergmixingangle,37,281structure,1,31self-gravitating,40,48,283,284Yang–Millssemisimplegroup,15,16,21equations,6,27simplerootvectors,21fields,28,29,303skew-symmetric,3,5,6model,5Sobolevinequality,58,104Yang–Mills–Higgssolitons,4,9equations,27spaceofmoduli,29theory,26,29