[Continuité hölderienne des solutions d'un problème de calcul des variations]
Pour un problème de calcul des variations multidimensionnel, où le lagrangien convexe ne dépend que du gradient, on montre que la continuité de la fonction ϕ définissant la condition de Dirichlet au bord implique la continuité des minimiseurs sur l'adhérence du domaine. Lorsque ϕ est lipschitzienne, alors les minimiseurs sont hölderiens.
For the basic problem in the calculus of variations where the Lagrangian is convex and depends only on the gradient, we establish the continuity of the solutions when the Dirichlet boundary condition is defined by a continuous function ϕ. When ϕ is Lipschitz continuous, then the solutions are Hölder continuous.
Accepté le :
Publié le :
Pierre Bousquet 1 ; Carlo Mariconda 2 ; Giulia Treu 2
@article{CRMATH_2008__346_23-24_1301_0,
author = {Pierre Bousquet and Carlo Mariconda and Giulia Treu},
title = {H\"older continuity of solutions to a basic problem in the calculus of variations},
journal = {Comptes Rendus. Math\'ematique},
pages = {1301--1305},
publisher = {Elsevier},
volume = {346},
number = {23-24},
year = {2008},
doi = {10.1016/j.crma.2008.10.001},
language = {en},
}
TY - JOUR AU - Pierre Bousquet AU - Carlo Mariconda AU - Giulia Treu TI - Hölder continuity of solutions to a basic problem in the calculus of variations JO - Comptes Rendus. Mathématique PY - 2008 SP - 1301 EP - 1305 VL - 346 IS - 23-24 PB - Elsevier DO - 10.1016/j.crma.2008.10.001 LA - en ID - CRMATH_2008__346_23-24_1301_0 ER -
%0 Journal Article %A Pierre Bousquet %A Carlo Mariconda %A Giulia Treu %T Hölder continuity of solutions to a basic problem in the calculus of variations %J Comptes Rendus. Mathématique %D 2008 %P 1301-1305 %V 346 %N 23-24 %I Elsevier %R 10.1016/j.crma.2008.10.001 %G en %F CRMATH_2008__346_23-24_1301_0
Pierre Bousquet; Carlo Mariconda; Giulia Treu. Hölder continuity of solutions to a basic problem in the calculus of variations. Comptes Rendus. Mathématique, Volume 346 (2008) no. 23-24, pp. 1301-1305. doi : 10.1016/j.crma.2008.10.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.10.001/
[1] On the lower bounded slope condition, J. Convex Anal., Volume 14 (2007) no. 1, pp. 119-136
[2] P. Bousquet, Boundary continuity of solutions to a basic problem in the calculus of variations, submitted for publication
[3] On the bounded slope condition and the validity of the Euler Lagrange equation, SIAM J. Control Optim., Volume 40 (2001/2002), pp. 1270-1279
[4] Comparison results and estimates on the gradient without strict convexity, SIAM J. Control Optim., Volume 46 (2007), pp. 738-749
[5] Continuity of solutions to a basic problem in the calculus of variations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 4 (2005), pp. 511-530
[6] Growth conditions and regularity, a counterexample, Manuscripta Math., Volume 59 (1987), pp. 245-248
[7] On the regularity of the minima of variational integrals, Acta Math., Volume 148 (1982), pp. 31-46
[8] Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003
[9] On the bounded slope condition, Pacific J. Math., Volume 18 (1966) no. 3, pp. 495-511
[10] Convex sets and the bounded slope condition, Pacific J. Math., Volume 25 (1968), pp. 511-522
[11] The absence of the continuity and Hölder continuity of the solutions of quasilinear elliptic equations near a nonregular boundary, Trans. Moscow Math. Soc., Volume 26 (1974), pp. 73-93
[12] Regularity for some scalar variational problems under general growth conditions, J. Optim. Theory Appl., Volume 90 (1996) no. 1, pp. 161-181
[13] Gradient maximum principle for minima, J. Optim. Theory Appl., Volume 112 (2002), pp. 167-186
[14] Existence and Lipschitz regularity for minima, Proc. Amer. Math. Soc., Volume 130 (2002) no. 2, pp. 395-404
[15] Lipschitz regularity for minima without strict convexity of the Lagrangian, J. Differential Equation, Volume 243 (2007), pp. 388-413
[16] C. Mariconda, G. Treu, Local Lipschitz regularity of minima for a scalar problem of the calculus of variations, Commun. Contemp. Math., in press
[17] C. Mariconda, G. Treu, Hölder regularity for a classical problem of the calculus of variations, submitted for publication
[18] Fine Regularity of Solutions of Elliptic Partial Differential Equations, Mathematical Surveys and Monographs, vol. 51, Amer. Math. Soc., 1997
[19] 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
[20] Simplified excision techniques for free discontinuity problems in several variables, J. Funct. Anal., Volume 151 (1997), pp. 1-34
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