Uniform Estimates for Dirichlet Problems in Perforated Domains

Abstract

This paper studies the Dirichlet problem for Laplace’s equation in a domain $$\varOmega _{\varepsilon , \eta }$$ perforated with small holes, where $$\varepsilon $$ represents the scale of the minimal distances between holes and $$\eta $$ 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 $$\eta $$ . We also show that these estimates are either optimal or near optimal.