On the existence of correctors for the stochastic homogenization of viscous hamilton-jacobi equations

Pierre Cardaliaguet (1) , Panagiotis E Souganidis (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Department of Mathematics [Chicago]

We prove, under some assumptions, the existence of correctors for the stochastic homoge-nization of 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 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 non convex Hamiltoni-ans and general stationary ergodic media.

Domaines: Equations aux dérivées partielles [math.AP]

Pierre Cardaliaguet, Panagiotis E Souganidis. On the existence of correctors for the stochastic homogenization of viscous hamilton-jacobi equations. Comptes rendus de l'Académie des sciences. Série I, Mathématique, 2017, 355, pp.786-794. ⟨10.1016/j.crma.2017.06.001⟩. ⟨hal-01513374⟩