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

Francis Clarke

Description

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

Sujets

• Clarke
• David Hilbert
• Lagrangian
• Dirichlet's principle
• Hilbert's problems

Informations

• Publié par: mijec
• Nombre de lectures: 13
• Langue: English

Extrait

February28,200812:5WSPC-ProceedingsTrimSize:9inx6inClarke˙Regularity˙Final Regularityofsolutionstoone-dimensionaland multi-dimensionalproblemsinthecalculusofvariations FrancisClarke InstitutuniversitairedeFranceetUniversite´deLyon InstitutCamilleJordanUMR5208 Universite´ClaudeBernardLyon1 LaDoua,69622Villeurbanne,France E-mail:[e-mail] Wereviewthelong-standingissueofregularityofsolutionstothebasicprob- leminthecalculusofvariations,inboththeone-dimensionalandthemulti- dimensionalsettings.Itisshownhowcertainrecentresultsfitinwiththe classicalones,inparticularthetheoriesofDeGiorgiandHilbert-Haar. Keywords:Calculusofvariations;regularity;necessaryconditions;existence. 1 1.Introduction Webegininthemiddle,withtwoofthecelebratedproblemsproposedby HilbertinParisin1900: The20thproblem:Isitnotthecasethateveryregularvariationalprob- lemhasasolution,providedcertainassumptionsontheboundarycondi- tionsaresatisfied,andprovidedalso,ifneedbe,thattheconceptofsolution issuitablyextended? The19thproblem:Arethesolutionsofregularproblemsinthecalculus ofvariationsalwaysanalytic? Thesequestionsbearuponthefollowingbasicprobleminthecalculus ofvariations:tominimizethefunctional Z J(u):=F(x,u(x),Du(x))dx Ω overthefunctionsu:Ω → R assumingprescribedvaluesonΓ:=∂Ω: u(x)=φ(x),x ∈ Γ.