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Comptes Rendus Mathématique
Volume 343, n° 3
pages 225-228 (août 2006)
Doi : 10.1016/j.crma.2006.06.001
Received : 16 January 2006 ;  accepted : 1 June 2006
Continuité lipschitzienne des solutions dʼun problème en calcul des variations
Lipschitzian continuity of solutions for a problem in the calculus of variations
 

Pierre Bousquet , Francis Clarke
Institut Camille-Jordan, Université Claude-Bernard Lyon 1, 69622 Villeurbanne, France 

Résumé

Dans cette Note, on décrit quelques développements récents sur la régularité des minimiseurs u de la fonctionnelle  , définie sur lʼensemble des fonctions   dont la trace sur est égale à une certaine fonction . Notre travail sʼinscrit dans la théorie de Hilbert-Haar mais on remplace la traditionnelle condition de pente bornée par une condition de pente minorée , moins restrictive que la précédente car elle est satisfaite dès que est la restriction à dʼune fonction convexe, voire semiconvexe. Sous cette nouvelle condition et des hypothèses de convexité sur F et , on montre que tout minimiseur u est localement lipschitzien dans , et dans certains cas, continu sur  . Pour citer cet article : P. Bousquet, F. Clarke, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

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Abstract

In this Note, we describe some recent developments concerning the regularity of the minimizers u of  , over the functions   that assume given boundary values on . The classical Hilbert-Haar theory derives regularity of u from an assumption on , the well-known bounded slope condition . Instead of this, we impose the less restrictive lower (or upper) bounded slope condition , which is satisfied if is the restriction to of a convex (or even semiconvex) function. Under this new assumption and some convexity hypotheses on F and , we show that any minimizer u is locally Lipschitz in . In some cases we are also able to assert that u is continuous on  . To cite this article: P. Bousquet, F. Clarke, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

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