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 $O(\epsilon )$ when the problem was anchored in the reference domain $\epsilon \omega$. 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 $O({\epsilon }^{1/2})$. 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 Calderón-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 $0<\epsilon \ll 1$. Appealing to this, we established Poincaré-Sobolev inequalities of local type on perforated domains. Some of results in the present literature are new even for related linear elliptic models.

arXiv comment45 pages. Compared to the previous version, this has been improved a lot, and we welcome comments on this job

References in corpus (6)

• Convergence Rates in L^2 for Elliptic Homogenization Problems
• Quantitative stochastic homogenization and large-scale regularity
• Higher-order pathwise theory of fluctuations in stochastic homogenization
• Quantitative Estimates on Periodic Homogenization of Nonlinear Elliptic Operators
• suboptimal error estimates for homogenization of linear elasticity systems on perforated domains
• Large-Scale Lipschitz Estimates for Elliptic Systems with Periodic High-Contrast Coefficients

Cited by in corpus (2)

• suboptimal error estimates for homogenization of linear elasticity systems on perforated domains
• Weighted Meyers estimates in perforated domains