February WSPC Proceedings Trim Size: 9in x 6in Clarke˙Regularity˙Final

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Niveau: Supérieur, Doctorat, Bac+8
February 28, 2008 12:5 WSPC - Proceedings Trim Size: 9in x 6in Clarke˙Regularity˙Final 1 Regularity of solutions to one-dimensional and multi-dimensional problems in the calculus of variations Francis Clarke Institut universitaire de France et Universite de Lyon Institut Camille Jordan UMR 5208 Universite Claude Bernard Lyon 1 La Doua, 69622 Villeurbanne, France E-mail: We review the long-standing issue of regularity of solutions to the basic prob- lem in the calculus of variations, in both the one-dimensional and the multi- dimensional settings. It is shown how certain recent results fit in with the classical ones, in particular the theories of De Giorgi and Hilbert-Haar. Keywords: Calculus of variations; regularity; necessary conditions; existence. 1. Introduction We begin in the middle, with two of the celebrated problems proposed by Hilbert in Paris in 1900: The 20th problem: Is it not the case that every regular variational prob- lem has a solution, provided certain assumptions on the boundary condi- tions are satisfied, and provided also, if need be, that the concept of solution is suitably extended? The 19th problem: Are the solutions of regular problems in the calculus of variations always analytic? These questions bear upon the following basic problem in the calculus of variations: to minimize the functional J(u) := ∫ ? F (x, u(x), Du(x)) dx over

  • hilbert

  • lagrangian

  • giorgi's theorem

  • full regularity

  • regularity properties

  • hilbert-haar approach

  • dirichlet principle

  • hilbert's problems

  • function ?

  • now becomes


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February28,200812:5WSPC-ProceedingsTrimSize:9inx6inClarke˙Regularity˙FinalRegularityofsolutionstoone-dimensionalandmulti-dimensionalproblemsinthecalculusofvariationsFrancisClarkeInstitutuniversitairedeFranceetUniversite´deLyonInstitutCamilleJordanUMR5208Universite´ClaudeBernardLyon1LaDoua,69622Villeurbanne,FranceE-mail:clarke@math.univ-lyon1.frWereviewthelong-standingissueofregularityofsolutionstothebasicprob-leminthecalculusofvariations,inboththeone-dimensionalandthemulti-dimensionalsettings.Itisshownhowcertainrecentresultsfitinwiththeclassicalones,inparticularthetheoriesofDeGiorgiandHilbert-Haar.Keywords:Calculusofvariations;regularity;necessaryconditions;existence.11.IntroductionWebegininthemiddle,withtwoofthecelebratedproblemsproposedbyHilbertinParisin1900:The20thproblem:Isitnotthecasethateveryregularvariationalprob-lemhasasolution,providedcertainassumptionsontheboundarycondi-tionsaresatisfied,andprovidedalso,ifneedbe,thattheconceptofsolutionissuitablyextended?The19thproblem:Arethesolutionsofregularproblemsinthecalculusofvariationsalwaysanalytic?Thesequestionsbearuponthefollowingbasicprobleminthecalculusofvariations:tominimizethefunctionalZJ(u):=F(x,u(x),Du(x))dxΩoverthefunctionsuRassumingprescribedvaluesonΓ:=Ω:u(x)=φ(x),xΓ.