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This paper was devoted to study the quantitative homogenization problems fornonlinear elliptic operators in perforated domains. We obtained a sharp errorestimate $O(\varepsilon)$ when the problem was anchored in the reference domain$\varepsilon\omega$. If concerning a bounded perforated domain, one will see abad influence from the boundary layers, which leads to the loss of theconvergence rate by $O(\varepsilon^{1/2})$. Equipped with the error estimates,we developed both interior and boundary Lipschitz estimates at large-scales. Asan application, we received the so-called quenched Calder\'on-Zygumundestimates by Shen's real arguments. To overcome some difficulties, we improvedthe 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 typeon perforated domains. Some of results in the present literature are new evenfor related linear elliptic models.

Quantitative estimates for homogenization of nonlinear elliptic operators in perforated domains