Dirichlet Problems in Perforated Domains

In this paper we establish W^1, p estimates for solutions u_ to Laplace's equation with the Dirichlet condition in a bounded and perforated, not necessarily periodically, C¹ domain, in Rᵈ. The bounding constants depend explicitly on two small parameters and, where represents the scale of the minimal distance between holes, and denotes the ratio between the size of the holes and. The proof relies on a large-scale Lᵖ estimate for u_, whose proof is divided into two parts. In the first part, we show that as, approach zero, harmonic functions in, may be approximated by solutions of an intermediate problem for a Schr\"odinger operator in. In the second part, a real-variable method is employed to establish the large-scale Lᵖ estimate for u_ by using the approximation at scales above. The results are sharp except in the case d 3 and p=d or d^.