Dirichlet problems in perforated domains

In this article, we establish W-1,W- p estimates for solutions u(epsilon) to Laplace's equation with the Dirichlet condition in a bounded and perforated, not necessarily periodically, C1 domain Omega(epsilon), eta in R-d. The bounding constants depend explicitly on two small parameters epsilon and eta, where epsilon represents the scale of the minimal distance between holes, and eta denotes the ratio between the size of the holes and epsilon. The proof relies on a large-scale L-p estimate for del u(epsilon), whose proof is divided into two parts. In the first part, we show that as epsilon, eta approach zero, harmonic functions in Omega(epsilon), eta may be approximated by solutions of an intermediate problem for a Schr & ouml;dinger operator in Omega. In the second part, a real-variable method is employed to establish the large-scale Lp estimate for del u(epsilon) by using the approximation at scales above epsilon. The results are shown to be sharp except in one particular case d >= 3 and p = d or d '.