Loss of quasiconvexity in the periodic homogenization of viscous Hamilton-Jacobi equations

TL;DR

This paper demonstrates that in periodic homogenization of viscous Hamilton-Jacobi equations, the effective Hamiltonian can lose quasiconvexity, contrasting with the first order case, and this loss is shown to be a generic phenomenon in higher dimensions.

Contribution

It reveals that quasiconvexity is not necessarily preserved in the homogenization of viscous Hamilton-Jacobi equations, unlike in the first order case, and shows this loss is generic in higher dimensions.

Findings

Effective Hamiltonian may not be quasiconvex after homogenization.

Loss of quasiconvexity is generic in higher dimensions.

Homogenization preserves quasiconvexity only in the first order case.

Abstract

We show that, in the periodic homogenization of uniformly elliptic Hamilton-Jacobi equations in any dimension, the effective Hamiltonian does not necessarily inherit the quasiconvexity property (in the momentum variables) of the original Hamiltonian. This observation is in sharp contrast with the first order case, where homogenization is known to preserve quasiconvexity. We also show that the loss of quasiconvexity is, in a way, generic: when the spatial dimension is $1$, every convex function $G$ can be modified on an arbitrarily small open interval so that the new function $\tilde{G}$ is quasiconvex and, for some $1$-periodic and Lipschitz continuous $V$, the effective Hamiltonian arising from the homogenization of the uniformly elliptic Hamilton-Jacobi equation with the Hamiltonian $H(p,x)=\tilde{G}(p)+V(x)$ is not quasiconvex.