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1、CriticaltoinPTheoryandApplicationstoNonlineartialDierenEquationsJeanwhinMatepartemenDdeematiqueMatheersitUnivCatholiquedeainLouvB-1348e,ain-la-NeuvLouvBelgiummawhin@math.ucl.ac.beLecturesengivattheVIGREMinicourseonariationalVMethodandNonlinearPDEyersitUnivofUtah,yMa2
2、8-June8,2002Chapter1troInductionalCriticointporytheisamodernolutionevofanoldpartofmathematicalanal-ysishwhiciscalledthealculuscofvariations.Problemsofthecalculusofari-vationscanbetracedtoGreekmathematics,forexamplewithDido'sproblemofndingtheclosedecurvofxedlengthsurrou
3、ndingamaximalarea.But,ifewexceptsomequestionsofopticstreatedybHeroofAlexandriaetPierredeermat,Ftherstproblemsofthefuturecalculusofariationsvwheretroinducedattheeryvbirthofthecalculus,attheendofthethteenensev.turyceneWcanrecallNewton'sproblemofthesolidofolutionrevoering
4、theleastresistanceina
uid,ortheproblemofndingtheshortestecurvjoiningowtxedptsoinintheplane,thespaceoronasurface(desicoge),ortheproblemofndingtheecurvalonghwhicafrictionlessparticleesvmoinaerticalvplane,underthesoleeectof,yvitgrafromoneptointoanother(onechroachystrbr)
5、.Intrastcontoclassicalproblemsofummaximandum,minimwhereonetriestolocalizetheptsoinathwhicarealfunctionofoneoreralsevariablesvhesreacaummaximoraumminimalue,vandhwhicevhaconstitutedastrongationmotivforthecreationofthecalculus,theeabvotionedmenproblemsrequirendingaecurv(i.
6、e.oneoreralsevfunctionsdeningthise)curvhwhicminimizesormaximizesaengivaluedreal-vexpressiondependinguponallthealuesvofthisorthosefunctions.Inmodernterms(andithasentakturiescentorealizeitandemakitt),ecientheproblemsofthecalculusofariationsvconsistinndingumsmaximorumsmi
7、nimofrealfunctionsdenedonafunctionspace.Theypicaltproblemofthecalculusofariations,vaswnshoybEulerinthemiddleofthethteeneigh,turycenconsistsinndingtheumminim(ortheum)maximofanexpressionoftheypteZb0f(x;y(x);y(x))dx;(1.1)aervoallrealtlysuciensmoothfunctionsyervo[a;b]taki
8、ngrespeectivxedaluesvAandBataandb:Euleredwshoinaquiteindirectyaw(reductiontoanite-dimensionalximation