SMM Self Dual Partial Differential Systems And Their Variational Principles (2009) Ghoussoub

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1、Self-dualPartialDifferentialSystemsandTheirVariationalPrinciplesNassifGhoussoubSelf-dualPartialDifferentialSystemsandTheirVariationalPrinciples123NassifGhoussoubUniversityofBritishColumbiaDepartmentofMathematicsVancouverBCV6T1Z2Canadanassif@math.ubc.caISSN:1439-7382ISBN:978-0-387-84896-

2、9e-ISBN:978-0-387-84897-6DOI:10.1007/978-0-387-84897-6LibraryofCongressControlNumber:2008938377MathematicsSubjectClassification(2000):46-xx,35-xxcSpringerScience+BusinessMedia,LLC2009Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenpermissionofthepubli

3、sher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnec-tionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhere

4、afterdevelopedisforbidden.Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjecttoproprietaryrights.Printedonacid-freepaperspringer.comToMireille.PrefaceHowtosolvepar

5、tialdifferentialsystemsbycompletingthesquare.Thiscouldwellhavebeenthetitleofthismonographasitgrewintoaprojecttodevelopasys-tematicapproachforassociatingsuitablenonnegativeenergyfunctionalstoalargeclassofpartialdifferentialequations(PDEs)andevolutionarysystems.Theminimaofthesefunctionals

6、aretobethesolutionsweseek,notbecausetheyarecriticalpoints(i.e.,fromthecorrespondingEuler-Lagrangeequations)butfromalsobe-ingzerosofthesefunctionals.TheapproachcanbetracedbacktoBogomolnyistrickofcompletingsquaresinthebasicequationsofquantumfieldtheory(e.g.,Yang-Mills,Seiberg-Witten,Ginzbu

7、rg-Landau,etc.,),whichallowsforthederiva-tionoftheso-calledself(orantiself)dualversionoftheseequations.Inreality,theself-dualLagrangiansweconsiderherewereinspiredbyavariationalap-proachproposedover30yearsagobyBrezisandEkelandfortheheatequation´andothergradientflowsofconvexenergi