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 $ω_{ε, η}$ perforated periodically with small holes in $\mathbb{R}^d$, where $ε$ represents the scale of the minimal distances between holes and $η$ the ratio between the scale of sizes of holes and $ε$. We establish $W^{1, p}$ estimates for solutions with bounding constants depending explicitly on $ε$ and $η$. The proof relies on a large-scale Lipschitz estimate for harmonic functions in perforated domains. The results are optimal for $d\ge 2$.

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