Comptes Rendus
no. 7
Partial differential equations
On the existence of correctors for the stochastic homogenization of viscous Hamilton–Jacobi equations
[Sur l'existence de correcteurs en homogénéisation stochastique d'équations de Hamilton–Jacobi]

Présenté par : Haim Brézis

Pierre Cardaliaguet1 ; Panagiotis E. Souganidis2
1 Ceremade, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France
2 Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 786-794.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.06.001
Pierre Cardaliaguet 1 ; Panagiotis E. Souganidis 2

1 Ceremade, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France
2 Department of Mathematics, University of Chicago, Chicago, IL 60637, USA 
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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/

[1] S.N. Armstrong; P.E. Souganidis Stochastic homogenization of Hamilton–Jacobi and degenerate Bellman equations in unbounded environments, J. Math. Pures Appl., Volume 97 (2012), pp. 460-504

[2] S.N. Armstrong; P.E. Souganidis Stochastic homogenization of level-set convex Hamilton–Jacobi equations, Int. Math. Res. Not., Volume 2013 (2013) no. 15, pp. 3420-3449

[3] S.N. Armstrong; P. Cardaliaguet Stochastic homogenization of quasilinear Hamilton–Jacobi equations and geometric motions, 2015 (arXiv preprint) | arXiv

[4] S.N. Armstrong; H.V. Tran Stochastic homogenization of viscous Hamilton–Jacobi equations and applications, Anal. PDE, Volume 7 (2015) no. 8, pp. 1969-2007

[5] S.N. Armstrong; H.V. Tran; Y. Yu Stochastic homogenization of nonconvex Hamilton–Jacobi equations in one space dimension, J. Differ. Equ., Volume 261 (2016) no. 5, pp. 2702-2737

[6] M.G. Crandall; H. Ishii; P.-L. Lions User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., Volume 27 (1992) no. 1, pp. 1-67

[7] M. Damron; J. Hanson Busemann functions and infinite geodesics in two-dimensional first-passage percolation, Commun. Math. Phys., Volume 325 (2014) no. 3, pp. 917-963

[8] A. Davini; A. Siconolfi Exact and approximate correctors for stochastic Hamiltonians: the 1-dimensional case, Math. Ann., Volume 345 (2009) no. 4, pp. 749-782

[9] W.M. Feldman; P.E. Souganidis Homogenization and non-homogenization of certain non-convex Hamilton–Jacobi equations, 2016 (arXiv preprint) | arXiv

[10] E. Kosygina; F. Rezakhanlou; S.R.S. Varadhan Stochastic homogenization of Hamilton–Jacobi–Bellman equations, Commun. Pure Appl. Math., Volume 59 (2006) no. 10, pp. 1489-1521

[11] C. Licea; C.M. Newman Geodesics in two-dimensional first-passage percolation, Ann. Probab., Volume 24 (1996) no. 1, pp. 399-410

[12] P.-L. Lions, G.C. Papanicolaou, S.R.S. Varadhan, Homogenization of Hamilton–Jacobi equations. Unpublished preprint, 1987.

[13] P.-L. Lions; P.E. Souganidis Correctors for the homogenization of Hamilton–Jacobi equations in the stationary ergodic setting, Commun. Pure Appl. Math., Volume 56 (2003) no. 10, pp. 1501-1524

[14] P.-L. Lions; P.E. Souganidis Homogenization of “viscous” Hamilton–Jacobi equations in stationary ergodic media, Commun. Partial Differ. Equ., Volume 30 (2005) no. 1–3, pp. 335-375

[15] P.-L. Lions; P.E. Souganidis Stochastic homogenization of Hamilton–Jacobi and “viscous” Hamilton–Jacobi equations with convex nonlinearities—revisited, Commun. Math. Sci., Volume 8 (2010) no. 2, pp. 627-637

[16] F. Rezakhanlou; J.E. Tarver Homogenization for stochastic Hamilton–Jacobi equations, Arch. Ration. Mech. Anal., Volume 151 (2000) no. 4, pp. 277-309

[17] P.E. Souganidis Stochastic homogenization of Hamilton–Jacobi equations and some applications, Asymptot. Anal., Volume 20 (1999) no. 1, pp. 1-11

[18] B. Ziliotto Stochastic homogenization of nonconvex Hamilton–Jacobi equations: a counterexample, Commun. Pure Appl. Math. (2017) (in press in press) | DOI

Cité par Sources :

☆ 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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