Rate of convergence for periodic homogenization of convex Hamilton-Jacobi equations in one dimension

TL;DR

This paper establishes the optimal convergence rate of order () for solutions of periodic convex Hamilton-Jacobi equations in one dimension, using control theory and ergodic theorems.

Contribution

It provides a simple proof of the () convergence rate for a broad class of convex Hamiltonians in one dimension, including classical mechanics cases.

Findings

Proves () convergence rate for viscosity solutions.

Uses optimal control and ergodic theory techniques.

Applicable to Hamiltonians with separable potentials.

Abstract

Let $u^\varepsilon$ and $u$ be viscosity solutions of the oscillatory Hamilton-Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence $\mathcal{O}(\varepsilon)$ of $u^\varepsilon \rightarrow u$ as $\varepsilon \rightarrow 0^+$ for a large class of convex Hamiltonians $H(x,y,p)$ in one dimension. This class includes the Hamiltonians from classical mechanics with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for periodic functions in dimension $n = 1$.