Boundary Korn Inequality and Neumann Problems in Homogenization of Systems of Elasticity

TL;DR

This paper develops uniform regularity estimates for Neumann problems in elasticity systems with oscillating coefficients, using a boundary Korn inequality and large-scale Rellich estimates to advance homogenization theory.

Contribution

It introduces a boundary Korn inequality for elasticity systems and applies it to establish optimal regularity estimates in homogenization of elliptic systems.

Findings

Established uniform regularity estimates for Neumann problems

Proved a boundary Korn inequality for elasticity systems

Utilized large-scale Rellich estimates for boundary analysis

Abstract

This paper concerns with a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients, arising in the theory of homogenization. We establish uniform optimal regularity estimates for solutions of Neumann problems in a bounded Lipschitz domain with $L^2$ boundary data. The proof relies on a boundary Korn inequality for solutions of systems of linear elasticity and uses a large-scale Rellich estimate obtained in \cite{Shen-2016}.