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

In this Note, we describe some recent developments concerning the regularity of the minimizers u of ΩF(u)+G(x,u), over the functions uW1,1(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 Ω¯.

Dans cette Note, on décrit quelques développements récents sur la régularité des minimiseurs u de la fonctionnelle ΩF(u)+G(x,u), définie sur l'ensemble des fonctions W1,1(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 Ω¯.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2006.06.001
Bousquet, Pierre 1; Clarke, Francis 1

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

[1] P. Bousquet, The lower bounded slope condition. J. Convex Anal., in press

[2] P. Bousquet, Local Lipschitz continuity of solutions of nonlinear elliptic pde's, submitted for publication

[3] P. Bousquet, F. Clarke, Local Lipschitz continuity of solutions to a problem in the calculus of variations, submitted for publication

[4] Cellina, A. On the bounded slope condition and the validity of the Euler Lagrange equation, SIAM J. Control Optim., Volume 40 (2001/2002) no. 4, pp. 1270-1279

[5] Clarke, F. Continuity of solutions to a basic problem in the calculus of variations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume 4 (2005) no. 3, pp. 511-530

[6] Giusti, E. Direct Methods in the Calculus of Variations, World Scientific Publishing Co. Inc., River Edge, NJ, 2003

[7] Hartman, P. On the bounded slope condition, Pacific J. Math., Volume 18 (1966), pp. 495-511

[8] Hartman, P. Convex sets and the bounded slope condition, Pacific J. Math., Volume 25 (1968), pp. 511-522

[9] Hartman, P.; Stampacchia, G. On some non-linear elliptic differential-functional equations, Acta Math., Volume 115 (1966), pp. 271-310

[10] Mariconda, C.; Treu, G. Gradient maximum principle for minima, J. Optim. Theory Appl., Volume 112 (2002) no. 1, pp. 167-186

[11] Mariconda, C.; Treu, G. Existence and Lipschitz regularity for minima, Proc. Amer. Math. Soc., Volume 130 (2002) no. 2, pp. 395-404

[12] Miranda, M. Un teorema di esistenza e unicità per il problema dell'area minima in n variabili, Ann. Scuola Norm. Sup. Pisa (3), Volume 19 (1965), pp. 233-249

[13] Stampacchia, G. On some regular multiple integral problems in the calculus of variations, Comm. Pure Appl. Math., Volume 16 (1963), pp. 383-421

Cited by Sources: