Uniform Estimates for Dirichlet Problems in Perforated Domains

This paper studies the Dirichlet problem for Laplace’s equation in a domain ${\mathrm{\Omega }}_{\epsilon ,\eta }$ perforated with small holes, 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 the small parameters $\epsilon$ and $\eta$. We also show that these estimates are either optimal or near optimal.