Dirichlet Problems in Perforated Domains

In this paper we establish W1,p estimates for solutions u_ε to Laplace's equation with the Dirichlet condition in a bounded and perforated, not necessarily periodically, C¹ domain Ωε, η in Rᵈ. The bounding constants depend explicitly on two small parameters ε and η, where ε represents the scale of the minimal distance between holes, and η denotes the ratio between the size of the holes and ε. The proof relies on a large-scale Lᵖ estimate for ∇ u_ε, whose proof is divided into two parts. In the first part, we show that as ε, η approach zero, harmonic functions in Ωε, η may be approximated by solutions of an intermediate problem for a Schr\"odinger operator in Ω. In the second part, a real-variable method is employed to establish the large-scale Lᵖ estimate for ∇ u_ε by using the approximation at scales above ε. The results are sharp except in the case d≥ 3 and $p=d$ or d^.