Dirichlet Problems in Perforated Domains

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