A Partial Homogenization Result for Nonconvex Viscous Hamilton-Jacobi Equations

TL;DR

This paper establishes homogenization results for nonconvex viscous Hamilton-Jacobi equations in a stationary ergodic setting, revealing conditions under which homogenization occurs and simplifying the problem to localized nonconvexity.

Contribution

It provides a general homogenization result for nonconvex viscous Hamilton-Jacobi equations and introduces a new proof approach for convex cases.

Findings

Homogenization occurs for a non-empty set within every level set of the effective Hamiltonian.

Homogenization is proven for the minimal level set of the effective Hamiltonian.

The problem reduces to cases where nonconvexity is localized in the gradient variable.

Abstract

We provide a general result concerning the homogenization of nonconvex viscous Hamilton-Jacobi equations in the stationary, ergodic setting. In particular, we show that homogenization occurs for a non-empty set of points within every level set of the effective Hamiltonian, and for every point in the minimal level set of the effective Hamiltonian. In addition, these methods provide a new proof of homogenization, in full, for convex equations and, for a class of level-set convex equations. Finally, we prove that the question of homogenization for first order equations reduces to the case that the nonconvexity of the Hamiltonian is localized in the gradient variable.