Homogenization and Non-Homogenization of certain Non-Convex Hamilton-Jacobi Equations

TL;DR

This paper investigates the conditions under which certain non-convex Hamilton-Jacobi equations in random media do or do not homogenize, revealing that local properties like saddle-points hinder homogenization, while specific geometric conditions promote it.

Contribution

It identifies two classes of non-convex Hamiltonians with distinct homogenization behaviors and introduces a new class with star-shaped sub-level sets that do homogenize.

Findings

Non-homogenization occurs for Hamiltonians with strict saddle-points.

Homogenization occurs for Hamiltonians with uniformly strictly star-shaped sub-level sets.

Counterexamples demonstrate the impact of local properties on homogenization behavior.

Abstract

We continue the study of the homogenization of coercive non-convex Hamilton-Jacobi equations in random media identifying two general classes of Hamiltonians with very distinct behavior. For the first class there is no homogenization in a particular environment while for the second homogenization takes place in environments with finite range dependence. Motivated by the recent counter-example of Ziliotto, who constructed a coercive but non-convex Hamilton-Jacobi equation with stationary ergodic random potential field for which homogenization does not hold, we show that same happens for coercive Hamiltonians which have a strict saddle-point, a very local property. We also identify, based on the recent work of Armstrong and Cardaliaguet on the homogenization of positively homogeneous random Hamiltonians in environments with finite range dependence, a new general class Hamiltonians, namely equations with uniformly strictly star-shaped sub-level sets, which homogenize.