Dirichlet Problems in Perforated Domains

Abstract

In this paper we establish $W^{1,p}$ estimates for solutions $u_{\epsilon }$ to Laplace's equation with the Dirichlet condition in a bounded and perforated, not necessarily periodically, $C^1$ domain ${\Omega }_{\epsilon ,\eta }$ in ${\mathbb{R}}^d$. 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 $\nabla u_{\epsilon }$, 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 Schrödinger operator in $\Omega$. In the second part, a real-variable method is employed to establish the large-scale $L^p$ estimate for $\nabla u_{\epsilon }$ by using the approximation at scales above $\epsilon$. The results are sharp except in the case $d\ge 3$ and $p=d$ or $d^{\prime }$.

37 pages