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

Let Formula: see text 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 Formula: see text of Formula: see text as Formula: see text for a large class of convex Hamiltonians Formula: see text 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 Formula: see text.