suboptimal error estimates for homogenization of linear elasticity systems on perforated domains

TL;DR

This paper derives near-optimal error estimates for linear elasticity in perforated domains, advancing homogenization theory with new techniques and sharper bounds.

Contribution

It introduces novel error estimates for elasticity systems in perforated domains using weighted inequalities and duality methods, improving upon previous results.

Findings

Established $L^{rac{2d}{d-1- au}}$-error estimates with $O(ig(rac{ au}{2}ig))$ rate

Derived $L^2$-error estimates with $O(rac{5}{6}} ext{log}^{2/3}(1/ig(rac{1}{ au}ig)))$ rate

Developed a new weighted extension theorem for perforated domains.

Abstract

In the present work, we established almost-sharp error estimates for linear elasticity systems in periodically perforated domains. The first result was $L^{\frac{2d}{d-1-\tau}}$-error estimates $O\big(\varepsilon^{1-\frac{\tau}{2}}\big)$ with $0<\tau<1$ for a bounded smooth domain. It followed from weighted Hardy-Sobolev's inequalities and a suboptimal error estimate for the square function of the first-order approximating corrector (which was earliest investigated by C. Kenig, F. Lin, Z. Shen \cite{KLS} under additional regularity assumption on coefficients). The new approach relied on the weighted quenched Calder\'on-Zygmund estimate (initially appeared in A. Gloria, S. Neukamm, F. Otto's work \cite{Gloria_Neukamm_Otto_2015} for a quantitative stochastic homogenization theory). The second effort was $L^2$-error estimates $O\big(\varepsilon^{\frac{5}{6}}\ln^{\frac{2}{3}}(1/\varepsilon)\big)$ for a Lipschitz domain, followed from a new duality scheme coupled with interpolation inequalities. Also, we developed a new weighted extension theorem for perforated domains, and a real method imposed by Z. Shen \cite{S3} played a fundamental role in the whole project.