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

TL;DR

This paper investigates the rate at which solutions to oscillatory Hamilton-Jacobi equations converge to their homogenized limits in a periodic setting, under convexity and coercivity assumptions on the Hamiltonian.

Contribution

It provides a quantitative convergence rate analysis for periodic homogenization of Hamilton-Jacobi equations with convex Hamiltonians.

Findings

Established explicit convergence rates for $u^\epsilon$ to $u$ as $\epsilon o 0$

Demonstrated the impact of convexity and coercivity on convergence speed

Extended previous qualitative results to quantitative estimates

Abstract

We study the rate of convergence of $u^\epsilon$, as $\epsilon \to 0+$, to $u$ in periodic homogenization of Hamilton-Jacobi equations. Here, $u^\epsilon$ and $u$ are viscosity solutions to the oscillatory Hamilton-Jacobi equation and its effective equation \begin{equation*} {\rm (C)_\epsilon} \qquad \begin{cases} u_t^\epsilon+H\left(\frac{x}{\epsilon},Du^\epsilon\right)=0 \qquad &\text{in} \ \mathbb{R}^n \times (0,\infty), u^\epsilon(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.