Rate of convergence for periodic homogenization of convex Hamilton–Jacobi equations in one dimension
Son N. T. Tu · Asymptotic Analysis · 2020
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].