Uniform Estimates for Dirichlet Problems in Perforated Domains

TL;DR

This paper derives explicit uniform estimates for solutions to Laplace's equation in perforated domains with small holes, analyzing how the estimates depend on the size and spacing of the holes, and demonstrating their optimality.

Contribution

It provides new explicit $W^{1,p}$ estimates for Dirichlet problems in perforated domains, detailing the dependence on small parameters and establishing near optimal bounds.

Findings

Derived explicit $W^{1,p}$ estimates depending on small parameters

Showed estimates are optimal or near optimal

Analyzed the influence of hole size and spacing on solution bounds

Abstract

This paper studies the Dirichlet problem for Laplace's equation in a domain $\Omega_{\varepsilon, \eta}$ perforated with small holes, where $\varepsilon$ represents the scale of the minimal distances between holes and $\eta$ the ratio between the scale of sizes of holes and $\varepsilon$. We establish $W^{1, p}$ estimates for solutions with bounding constants depending explicitly on the small parameters $\varepsilon$ and $\eta$. We also show that these estimates are either optimal or near optimal.