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

Abstract

Let ${\mathit{u}}^{\mathit{\epsilon }}$ 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 $\mathcal{O}\mathrm{(}\mathit{\epsilon }\mathrm{)}$ of ${\mathit{u}}^{\mathit{\epsilon }}\to \mathit{u}$ as $\mathit{\epsilon }\to 0^{+}$ for a large class of convex Hamiltonians $\mathit{H}\mathrm{(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathit{p}\mathrm{)}$ 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 $\mathit{n}=1$ .