Large-Scale Lipschitz Estimates for Elliptic Systems with Periodic High-Contrast Coefficients

TL;DR

This paper establishes uniform large-scale Lipschitz estimates for elliptic systems with periodic high-contrast coefficients, covering various inclusion cases and enabling explicit bounds for fundamental solutions.

Contribution

It provides the first uniform large-scale Lipschitz estimates for elliptic systems with periodic high-contrast coefficients, including soft and stiff inclusions.

Findings

Uniform Lipschitz estimates independent of contrast ratio

Explicit bounds for fundamental solutions and derivatives

Coverage of soft and stiff inclusion cases

Abstract

This paper is concerned with the large-scale regularity in the homogenization of elliptic systems of elasticity with periodic high-contrast coefficients. We obtain the large-scale Lipschitz estimate that is uniform with respect to the contrast ratio $\delta^2$ for $0<\delta<\infty$. Our study also covers the case of soft inclusions ($\delta=0$) as well as the case of stiff inclusions ($\delta=\infty$). The large-scale Lipschitz estimate, together with classical local estimates, allows us to establish explicit bounds for the matrix of fundamental solutions and its derivatives.