Dans cette Note, on décrit quelques développements récents sur la régularité des minimiseurs u de la fonctionnelle , définie sur l'ensemble des fonctions 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 .
In this Note, we describe some recent developments concerning the regularity of the minimizers u of , over the functions 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 .
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@article{CRMATH_2006__343_3_225_0,
author = {Bousquet, Pierre and Clarke, Francis},
title = {Continuit\'e lipschitzienne des solutions d'un probl\`eme en calcul des variations},
journal = {Comptes Rendus. Math\'ematique},
pages = {225--228},
publisher = {Elsevier},
volume = {343},
number = {3},
year = {2006},
doi = {10.1016/j.crma.2006.06.001},
language = {fr},
url = {http://www.numdam.org/articles/10.1016/j.crma.2006.06.001/}
}
TY - JOUR AU - Bousquet, Pierre AU - Clarke, Francis TI - Continuité lipschitzienne des solutions d'un problème en calcul des variations JO - Comptes Rendus. Mathématique PY - 2006 SP - 225 EP - 228 VL - 343 IS - 3 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2006.06.001/ DO - 10.1016/j.crma.2006.06.001 LA - fr ID - CRMATH_2006__343_3_225_0 ER -
%0 Journal Article %A Bousquet, Pierre %A Clarke, Francis %T Continuité lipschitzienne des solutions d'un problème en calcul des variations %J Comptes Rendus. Mathématique %D 2006 %P 225-228 %V 343 %N 3 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2006.06.001/ %R 10.1016/j.crma.2006.06.001 %G fr %F CRMATH_2006__343_3_225_0
Bousquet, Pierre; Clarke, Francis. Continuité lipschitzienne des solutions d'un problème en calcul des variations. Comptes Rendus. Mathématique, Tome 343 (2006) no. 3, pp. 225-228. doi : 10.1016/j.crma.2006.06.001. http://www.numdam.org/articles/10.1016/j.crma.2006.06.001/
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