An abstract Nash–Moser theorem with parameters and applications to PDEs.pdf

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1、Ann.I.H.PoincaréAN27(2010)377399www.elsevier.com/locate/anihpcAnabstractNashMosertheoremwithparametersandapplicationstoPDEsM.Bertia,∗,1,P.Bolleb,M.Procesia,1aDipartimentodiMatematicaeApplicazioniR.Caccioppoli,UniversitàdegliStudiNapoliFedericoII,ViaCin

2、tia,MonteS.Angelo,I-80126Napoli,ItalybUniversitédAvignonetdesPaysdeVaucluse,LaboratoiredAnalysenonLinéaireetGéométrie(EA2151),F-84018Avignon,FranceReceived26February2009;receivedinrevisedform8November2009;accepted8November2009Availableonline3December20

3、09AbstractWeproveanabstractNashMoserimplicitfunctiontheoremwithparameterswhichcoverstheapplicationstotheexistenceoffinitedimensional,differentiable,invarianttoriofHamiltonianPDEswithmerelydifferentiablenonlinearities.Themainnewfeatureoftheabstractiterat

4、iveschemeisthatthelinearizedoperators,inaneighborhoodoftheexpectedsolution,areinvertible,andsatisfythetameestimates,onlyforpropersubsetsoftheparameters.AsanapplicationweshowtheexistenceofperiodicsolutionsofnonlinearwaveequationsonRiemannianZollmanifold

5、s.Apointofinterestisthat,inpresenceofpossiblyverylargeclustersofsmalldivisors,duetoresonancephenomena,itismorenaturaltoexpectsolutionswithonlySobolevregularity.©2009ElsevierMassonSAS.Allrightsreserved.1.Introduction1.1.SmalldivisorsproblemsinHamiltonia

6、nPDEsBifurcationproblemsofperiodicandquasi-periodicsolutionsforHamiltonianPDEsarenaturallyaffectedbysmalldivisorsdifficulties:thestandardimplicitfunctiontheoremcannotbeappliedbecausethelinearizedoperatorshaveanunboundedinverse,duetoarbitrarilysmalldivis

7、orsintheirFourierseriesexpansions.ThisproblemhasbeenhandledforPDEswithanalyticnonlinearitiesviaKAMmethods,seee.g.Kuksin[22,23],Wayne[29],Pöschel[27],EliassonandKuksin[15],orviaNewton-typeiterativeschemesasdevelopedinCraigandWayne[14]andBourgain[710].Th

8、epioneeringKAMresultsin[22,29,27]werelimitedto1-dimensionalPDEs,withDirichletboundarycondi-tions,becausetheyrequiredtheeigenvaluesoftheLaplaciantobesimple(thesquarerootsoftheeigenvaluesarethenormalmodefrequenciesofsmalloscillations).Int