Convergence rates for general elliptic homogenization problems in a bounded Lipschitz domain

TL;DR

This paper advances the understanding of elliptic homogenization in Lipschitz domains by establishing new weighted estimates and sharp convergence rates, extending prior results to more general operators with less smooth coefficients.

Contribution

It introduces new weighted estimates for smoothing operators and achieves sharp $O( ext{epsilon})$ convergence rates in $L^p$, broadening previous homogenization results to more general elliptic problems.

Findings

Established new weighted estimates for smoothing operators.

Proved sharp $O( ext{epsilon})$ convergence rates in $L^p$ spaces.

Extended homogenization results to operators with lower order terms and nonsmooth coefficients.

Abstract

The paper extends the results obtained by C. Kenig, F. Lin and Z. Shen in \cite{SZW2} to more general elliptic homogenization problems in two perspectives: lower order terms in the operator and no smoothness on the coefficients. We do not repeat their arguments. Instead we find the new weighted-type estimates for the smoothing operator at scale $\varepsilon$, and combining some techniques developed by Z. Shen in \cite{SZW12} leads to our main results. In addition, we also obtain sharp $O(\varepsilon)$ convergence rates in $L^{p}$ with $p=2d/(d-1)$, which were originally established by Z. Shen for elasticity systems in \cite{SZW12}. Also, this work may be regarded as the extension of \cite{TS,TS2} developed by T. Suslina concerned with the bounded Lipschitz domain.