Quantitative homogenization of Hamilton--Jacobi equations on perforated domains with Dirichlet boundary conditions

TL;DR

This paper establishes an optimal convergence rate of order epsilon for the homogenization of convex Hamilton-Jacobi equations on perforated domains with Dirichlet boundary conditions, using control representations and metric properties.

Contribution

It introduces a novel analysis of the optimal control framework to achieve the first known optimal convergence rate in this setting.

Findings

Proves an $ ext{O}( ext{epsilon})$ convergence rate for homogenization.

Analyzes the metric function related to the control problem.

Handles singularities when optimal paths are time-limited.

Abstract

We study the periodic homogenization of convex Hamilton-Jacobi equations on perforated domains with Dirichlet boundary conditions. By analyzing the optimal control representation of the solutions and the properties of the metric function associated with the running cost, we establish the optimal convergence rate $\mathcal{O}(\varepsilon)$ for homogenization. A key aspect of our approach is the treatment of the singularity that arises when the optimal path does not fully utilize the available time.