[Sur l'existence de correcteurs en homogénéisation stochastique d'équations de Hamilton–Jacobi]
We prove, under some assumptions, the existence of correctors for the stochastic homogenization of “viscous” possibly degenerate Hamilton–Jacobi equations in stationary ergodic media. The general claim is that, assuming knowledge of homogenization in probability, correctors exist for all extreme points of the convex hull of the sublevel sets of the effective Hamiltonian. Even when homogenization is not a priori known, the arguments imply the existence of correctors and, hence, homogenization in some new settings. These include positively homogeneous Hamiltonians and, hence, geometric-type equations including motion by mean curvature, in radially symmetric environments and for all directions. Correctors also exist and, hence, homogenization holds for many directions for nonconvex Hamiltonians and general stationary ergodic media.
Nous démontrons l'existence, sous certaines conditions, de correcteurs en homogénéisation stochastique d'équations de Hamilton–Jacobi et d'équations de Hamilton–Jacobi visqueuses. L'énoncé général est que, si l'on sait qu'il y a homogénéisation en probabilité, un correcteur existe pour toute direction étant un point extrémal de l'enveloppe convexe d'un ensemble de niveau du Hamiltonien effectif. Même lorsque que l'homogénéisation n'est pas connue a priori, les arguments développés dans cette note montrent l'existence d'un correcteur, et donc l'homogénéisation, dans certains contextes. Cela inclut les équations de type géométrique dans des environnements dont la loi est à symmétrie radiale. Dans le cas général stationnaire ergodique et sans hypothèse de convexité sur le hamiltonien, on montre que des correcteurs existent pour plusieurs directions.
Accepté le :
Publié le :
Pierre Cardaliaguet 1 ; Panagiotis E. Souganidis 2
@article{CRMATH_2017__355_7_786_0,
author = {Pierre Cardaliaguet and Panagiotis E. Souganidis},
title = {On the existence of correctors for the stochastic homogenization of viscous {Hamilton{\textendash}Jacobi} equations},
journal = {Comptes Rendus. Math\'ematique},
pages = {786--794},
publisher = {Elsevier},
volume = {355},
number = {7},
year = {2017},
doi = {10.1016/j.crma.2017.06.001},
language = {en},
}
TY - JOUR AU - Pierre Cardaliaguet AU - Panagiotis E. Souganidis TI - On the existence of correctors for the stochastic homogenization of viscous Hamilton–Jacobi equations JO - Comptes Rendus. Mathématique PY - 2017 SP - 786 EP - 794 VL - 355 IS - 7 PB - Elsevier DO - 10.1016/j.crma.2017.06.001 LA - en ID - CRMATH_2017__355_7_786_0 ER -
%0 Journal Article %A Pierre Cardaliaguet %A Panagiotis E. Souganidis %T On the existence of correctors for the stochastic homogenization of viscous Hamilton–Jacobi equations %J Comptes Rendus. Mathématique %D 2017 %P 786-794 %V 355 %N 7 %I Elsevier %R 10.1016/j.crma.2017.06.001 %G en %F CRMATH_2017__355_7_786_0
Pierre Cardaliaguet; Panagiotis E. Souganidis. On the existence of correctors for the stochastic homogenization of viscous Hamilton–Jacobi equations. Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 786-794. doi : 10.1016/j.crma.2017.06.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.06.001/
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Cité par 13 documents. Sources : Crossref, zbMATH
☆ P. Cardaliaguet was partially supported by the ANR (Agence nationale de la recherche) project ANR-12-BS01-0008-01. P. E. Souganidis was partially supported by the National Science Foundation Grants DMS-1266383 and DMS-1600129 and the Office of Naval Research Grant N000141712095.
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