A periodic homogenization problem with defects rare at infinity

TL;DR

This paper studies a homogenization problem for diffusion equations with coefficients that are mostly periodic but have rare defects at infinity, establishing existence of correctors, identifying limits, and analyzing convergence rates.

Contribution

It introduces a novel homogenization framework for coefficients with non-local, rare defects at infinity, extending classical periodic homogenization theory.

Findings

Existence of a corrector for the problem

Identification of the homogenized limit

Analysis of convergence rates of solutions

Abstract

We consider a homogenization problem for the diffusion equation $-\operatorname{div}\left(a_{\varepsilon} \nabla u_{\varepsilon} \right) = f$ when the coefficient $a_{\varepsilon}$ is a non-local perturbation of a periodic coefficient. The perturbation does not vanish but becomes rare at infinity in a sense made precise in the text. We prove the existence of a corrector, identify the homogenized limit and study the convergence rates of $u_{\varepsilon}$ to its homogenized limit.