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

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.