Quantitative estimates for homogenization of nonlinear elliptic operators in perforated domains

Abstract

This paper was devoted to study the quantitative homogenization problems for nonlinear elliptic operators in perforated domains. We obtained a sharp error estimate $O(\varepsilon)$ when the problem was anchored in the reference domain $\varepsilon\omega$. If concerning a bounded perforated domain, one will see a bad influence from the boundary layers, which leads to the loss of the convergence rate by $O(\varepsilon^{1/2})$. Equipped with the error estimates, we developed both interior and boundary Lipschitz estimates at large-scales. As an application, we received the so-called quenched Calder\'on-Zygumund estimates by Shen's real arguments. To overcome some difficulties, we improved the extension theory from (\cite[Theorem 4.3]{OSY}) to $L^p$-versions with $\frac{2d}{d+1}-\epsilon<p<\frac{2d}{d-1}+\epsilon$ and $0<\epsilon\ll1$. Appealing to this, we established Poincar\'e-Sobolev inequalities of local type on perforated domains. Some of results in the present literature are new even for related linear elliptic models.