Quantitative homogenization of convex Hamilton-Jacobi equations with Neumann type boundary conditions

TL;DR

This paper develops a new approach to homogenize convex Hamilton-Jacobi equations with Neumann boundary conditions on perforated domains, achieving an optimal convergence rate of O(ε).

Contribution

It introduces a novel representation formula using the Skorokhod problem and modified Lagrangians, enabling precise homogenization analysis with Neumann boundary conditions.

Findings

Established a new representation formula for solutions.

Proved sub and superadditivity of extended metric functions.

Achieved the optimal convergence rate of O(ε) for homogenization.

Abstract

We study the periodic homogenization for convex Hamilton-Jacobi equations on perforated domains under the Neumann type boundary conditions. We consider two types of conditions, the oblique derivative boundary condition and the prescribed contact angle boundary condition, which is important in the front propagation. We first establish a new representation formula for the solution by using the Skorokhod problem and modified Lagrangians. By using this formula essentially, we prove the sub and superadditivity properties of the extended metric functions, which will be applied to obtain the optimal convergence rate $O(\varepsilon)$ for homogenization of Neumann type problems.