Uniform W^{1,p} Estimates for Systems of Linear Elasticity in a Periodic Medium

TL;DR

This paper establishes uniform W^{1,p} estimates for solutions to elliptic systems of linear elasticity with periodic coefficients, valid in Lipschitz domains and within specific p-ranges, which are proven to be sharp in low dimensions.

Contribution

It provides the first uniform W^{1,p} estimates for such elasticity systems with oscillating periodic coefficients, extending the understanding of regularity in these models.

Findings

Uniform W^{1,p} estimates obtained for solutions in Lipschitz domains.

p-ranges are sharp for dimensions 2 and 3.

Estimates hold for a broad class of elliptic systems with periodic coefficients.

Abstract

Let $\mathcal{L}_\epsilon$ be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients. We obtain the uniform $W^{1,p}$ estimate in a Lipschitz domain for solutions to the Dirichlet problem, where $(2n/(n+1)) -\delta<p<(2n/(n-1))+\delta$. The ranges of $p$'s are sharp for $n=2$ or 3.