Uniform $W^{1, p}$ Estimates and Large-Scale Regularity for Dirichlet Problems in Perforated Domains

In this paper we study the Dirichlet problem for Laplace's equation in a domain $\omega_{\epsilon, \eta}$ perforated periodically with small holes in $\mathbb{R}^d$, where $\epsilon$ represents the scale of the minimal distances between holes and $\eta$ the ratio between the scale of sizes of holes and $\epsilon$. We establish $W^{1, p}$ estimates for solutions with bounding constants depending explicitly on $\epsilon$ and $\eta$. The proof relies on a large-scale Lipschitz estimate for harmonic functions in perforated domains. The results are optimal for $d\ge 2$.