Convergence rate and uniform Lipschitz estimate in periodic homogenization of high-contrast elliptic systems

TL;DR

This paper investigates the convergence rates and uniform Lipschitz estimates in the periodic homogenization of high-contrast elliptic systems, providing a unified approach for various contrast regimes and periodicities.

Contribution

It introduces a unified method to quantify convergence rates and derive uniform regularity estimates in high-contrast elliptic homogenization.

Findings

Quantified convergence rates as periodicity tends to zero.

Derived uniform Lipschitz estimates independent of contrast.

Applicable to both zero and infinite contrast regimes.

Abstract

We consider the Dirichlet problem for elliptic systems with periodically distributed inclusions whose conduction parameter exhibits a significant contrast compared to the background media. We develop a unified method to quantify the convergence rates both as the periodicity of inclusions tends to zero and as the parameter approaches either zero or infinity. Based on the obtained convergence rates and a Campanato-type scheme, we also derive the regularity estimates that are uniform both in the periodicity and the contrast.