critical point theory and applications to nonlinear differential equations

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1、CriticaltoinPTheoryandApplicationstoNonlineartialDierenEquationsJeanwhinMatepartemenDdeematiqueMatheersitUnivCatholiquedeainLouvB-1348e,ain-la-NeuvLouvBelgiummawhin@math.ucl.ac.beLecturesengivattheVIGREMinicourseonariationalVMethodandNonlinearPDEyersitUnivofUta

2、h,yMa28-June8,2002Chapter1troInductionalCriticointporytheisamodernolutionevofanoldpartofmathematicalanal-ysishwhiciscalledthealculuscofvariations.Problemsofthecalculusofari-vationscanbetracedtoGreekmathematics,forexamplewithDido'sproblemofndingtheclosedecurvofxed

3、lengthsurroundingamaximalarea.But,ifewexceptsomequestionsofopticstreatedybHeroofAlexandriaetPierredeermat,Ftherstproblemsofthefuturecalculusofariationsvwheretroinducedattheeryvbirthofthecalculus,attheendofthethteenensev.turyceneWcanrecallNewton'sproblemofthesolido

4、folutionrevoeringtheleastresistanceina uid,ortheproblemofndingtheshortestecurvjoiningowtxedptsoinintheplane,thespaceoronasurface(desicoge),ortheproblemofndingtheecurvalonghwhicafrictionlessparticleesvmoinaerticalvplane,underthesoleeectof,yvitgrafromoneptointoa

5、nother(onechroachystrbr).Intrastcontoclassicalproblemsofummaximandum,minimwhereonetriestolocalizetheptsoinathwhicarealfunctionofoneoreralsevariablesvhesreacaummaximoraumminimalue,vandhwhicevhaconstitutedastrongationmotivforthecreationofthecalculus,theeabvotionedmen

6、problemsrequirendingaecurv(i.e.oneoreralsevfunctionsdeningthise)curvhwhicminimizesormaximizesaengivaluedreal-vexpressiondependinguponallthealuesvofthisorthosefunctions.Inmodernterms(andithasentakturiescentorealizeitandemakitt),ecientheproblemsofthecalculusofaria

7、tionsvconsistinndingumsmaximorumsminimofrealfunctionsdenedonafunctionspace.Theypicaltproblemofthecalculusofariations,vaswnshoybEulerinthemiddleofthethteeneigh,turycenconsistsinndingtheumminim(ortheum)maximofanexpressionoftheypteZb0f(x;y(x);y(x))dx;(1.1)aervoallr

8、ealtlysuciensmoothfunctionsyervo[a;b]takingrespeectivxedaluesvAandBataandb:Euleredwshoinaquiteindirectyaw(reductiontoanite-dimensionalximation