Continuité lipschitzienne des solutions d'un problème en calcul des variations

Pierre Bousquet; Francis Clarke

doi:10.1016/j.crma.2006.06.001

Calcul des variations

Continuité lipschitzienne des solutions d'un problème en calcul des variations
[Lipschitzian continuity of solutions for a problem in the calculus of variations]

Presented by: Haïm Brezis

Pierre Bousquet ¹; Francis Clarke ¹

¹ Institut Camille-Jordan, Université Claude-Bernard Lyon 1, 69622 Villeurbanne, France

Comptes Rendus. Mathématique, Volume 343 (2006) no. 3, pp. 225-228.

Abstract
Résumé

In this Note, we describe some recent developments concerning the regularity of the minimizers u of $\int_{Ω} F (\nabla u) + G (x, u)$ , over the functions $u \in W^{1, 1} (Ω)$ that assume given boundary values ϕ on ∂Ω. The classical Hilbert–Haar theory derives regularity of u from an assumption on ϕ, the well-known bounded slope condition. Instead of this, we impose the less restrictive lower (or upper) bounded slope condition, which is satisfied if ϕ is the restriction to ∂Ω of a convex (or even semiconvex) function. Under this new assumption and some convexity hypotheses on F and Ω, we show that any minimizer u is locally Lipschitz in Ω. In some cases we are also able to assert that u is continuous on $\bar{Ω}$ .

Dans cette Note, on décrit quelques développements récents sur la régularité des minimiseurs u de la fonctionnelle $\int_{Ω} F (\nabla u) + G (x, u)$ , définie sur l'ensemble des fonctions $W^{1, 1} (Ω)$ dont la trace sur ∂Ω est égale à une certaine fonction ϕ. Notre travail s'inscrit 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 ϕ est la restriction à ∂Ω d'une fonction convexe, voire semiconvexe. Sous cette nouvelle condition et des hypothèses de convexité sur F et Ω, on montre que tout minimiseur u est localement lipschitzien dans Ω, et dans certains cas, continu sur $\bar{Ω}$ .

Received: 2006-01-16
Accepted: 2006-06-01
Published online: 2006-07-05

DOI: 10.1016/j.crma.2006.06.001

Author's affiliations:

Pierre Bousquet ¹; Francis Clarke ¹

¹ Institut Camille-Jordan, Université Claude-Bernard Lyon 1, 69622 Villeurbanne, France

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     author = {Pierre Bousquet and Francis Clarke},
     title = {Continuit\'e lipschitzienne des solutions d'un probl\`eme en calcul des variations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {225--228},
     publisher = {Elsevier},
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Pierre Bousquet; Francis Clarke. Continuité lipschitzienne des solutions d'un problème en calcul des variations. Comptes Rendus. Mathématique, Volume 343 (2006) no. 3, pp. 225-228. doi : 10.1016/j.crma.2006.06.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.06.001/

References
Cited by

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