Quantitative estimates for homogenization of nonlinear elliptic operators in perforated domains (2001.06317v2)

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.