critical point theory and applications to nonlinear differential equations (1)

critical point theory and applications to nonlinear differential equations (1)

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时间:2019-08-06

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2、8-June8,2002Chapter1troInductionalCriticointporytheisamodernolutionevofanoldpartofmathematicalanal-ysishwhiciscalledthealculuscofvariations.Problemsofthecalculusofari-vationscanbetracedtoGreekmathematics,forexamplewithDido'sproblemof ndingtheclosedecurvof xedlengthsurrou

3、ndingamaximalarea.But,ifewexceptsomequestionsofopticstreatedybHeroofAlexandriaetPierredeermat,Fthe rstproblemsofthefuturecalculusofariationsvwheretroinducedattheeryvbirthofthecalculus,attheendofthethteenensev.turyceneWcanrecallNewton'sproblemofthesolidofolutionrevo ering

4、theleastresistanceina uid,ortheproblemof ndingtheshortestecurvjoiningowt xedptsoinintheplane,thespaceoronasurface(desicoge),ortheproblemof ndingtheecurvalonghwhicafrictionlessparticleesvmoinaerticalvplane,underthesolee ectof,yvitgrafromoneptointoanother(onechroachystrbr)

5、.Intrastcontoclassicalproblemsofummaximandum,minimwhereonetriestolocalizetheptsoinathwhicarealfunctionofoneoreralsevariablesvhesreacaummaximoraumminimalue,vandhwhicevhaconstitutedastrongationmotivforthecreationofthecalculus,theeabvotionedmenproblemsrequire ndingaecurv(i.

6、e.oneoreralsevfunctionsde ningthise)curvhwhicminimizesormaximizesaengivaluedreal-vexpressiondependinguponallthealuesvofthisorthosefunctions.Inmodernterms(andithasentakturiescentorealizeitandemakitt),ecientheproblemsofthecalculusofariationsvconsistin ndingumsmaximorumsmi

7、nimofrealfunctionsde nedonafunctionspace.Theypicaltproblemofthecalculusofariations,vaswnshoybEulerinthemiddleofthethteeneigh,turycenconsistsin ndingtheumminim(ortheum)maximofanexpressionoftheypteZb0f(x;y(x);y(x))dx;(1.1)aervoallrealtlysuciensmoothfunctionsyervo[a;b]taki

8、ngrespeectiv xedaluesvAandBataandb:Euleredwshoinaquiteindirectyaw(reductiontoa nite-dimensionalximation

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