Uniform Estimates for Dirichlet Problems in Perforated Domains

This paper studies the Dirichlet problem for Laplace's equation in a domain $Ω_{\varepsilon, η}$ perforated with small holes, where $\varepsilon$ represents the scale of the minimal distances between holes and $η$ the ratio between the scale of sizes of holes and $\varepsilon$. We establish $W^{1, p}$ estimates for solutions with bounding constants depending explicitly on the small parameters $\varepsilon$ and $η$. We also show that these estimates are either optimal or near optimal.

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