Slow periodic homogenization for Hamilton-Jacobi equations

TL;DR

This paper explores the rate of periodic homogenization for coercive Hamilton-Jacobi equations, demonstrating that nonconvex Hamiltonians can have a slower rate of (/2) compared to the convex case.

Contribution

It constructs examples of coercive nonconvex Hamiltonians with a homogenization rate of (), extending previous results that focused on convex Hamiltonians.

Findings

Nonconvex Hamiltonians can have a homogenization rate of ()

Previous rate for convex Hamiltonians is ()

Provides explicit examples illustrating the slower rate

Abstract

Capuzzo-Dolcetta and Ishii proved that the rate of periodic homogenization for coercive Hamilton-Jacobi equations is $O(\varepsilon^{1/3})$. We complement this result by constructing examples of coercive nonconvex Hamiltonians whose rate of periodic homogenization is $\Omega(\varepsilon^{1/2})$.