Rate of convergence in periodic homogenization of Hamilton-Jacobi equations: the convex setting

Abstract

We study the rate of convergence of $u^ε$, as $ε\to 0+$, to $u$ in periodic homogenization of Hamilton-Jacobi equations. Here, $u^ε$ and $u$ are viscosity solutions to the oscillatory Hamilton-Jacobi equation and its effective equation \begin{equation*} {\rm (C)_ε} \qquad \begin{cases} u_t^ε+H\left(\frac{x}ε,Du^ε\right)=0 \qquad &\text{in} \ \mathbb{R}^n \times (0,\infty), u^ε(x,0)=g(x) \qquad &\text{on} \ \mathbb{R}^n, \end{cases} \end{equation*} and \begin{equation*} {\rm (C)} \qquad \begin{cases} u_t+\overline{H}\left(Du\right)=0 \qquad &\text{in} \ \mathbb{R}^n \times (0,\infty), u(x,0)=g(x) \qquad &\text{on} \ \mathbb{R}^n, \end{cases} \end{equation*} respectively. We assume that the Hamiltonian $H=H(y,p)$ is coercive and convex in the $p$ variable and is $\mathbb{Z}^n$-periodic in the $y$ variable, and the initial data $g$ is bounded and Lipschitz continuous.