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

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