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

Abstract

Let $u^{\epsilon }$ 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 $\mathcal{O}(\epsilon )$ of $u^{\epsilon }\to u$ as $\epsilon \to 0^{+}$ 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 $n=1$.

23 pages, fixed an error in the statement of Lemma 2.8 from the previous version (results are unaffected)