Homogenization of some periodic Hamilton-Jacobi equations with defects

TL;DR

This paper investigates the homogenization process for certain stationary Hamilton-Jacobi equations with localized perturbations, revealing a limiting problem that combines periodic homogenization outside the origin with a specialized Dirichlet condition at the origin.

Contribution

It introduces a novel homogenization framework for Hamilton-Jacobi equations with localized defects, characterizing the effective behavior and boundary conditions at the defect point.

Findings

The limiting problem retains the effective Hamiltonian away from the defect.

A new effective Dirichlet condition at the defect point captures the perturbation.

The approach extends to various perturbation types and provides insights into defect effects.

Abstract

We study homogenization for a class of stationnary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, supplemented at the origin with an effective Dirichlet condition that keeps track of the perturbation. Various comments and extensions are discussed.