On uniform estimates for Laplace equation in balls with small holes

TL;DR

This paper investigates uniform estimates for the Laplace equation in three-dimensional balls with small holes, identifying the ranges of p for which these estimates hold or fail, with implications for homogenization problems.

Contribution

It provides a near-complete characterization of uniform $W^{1,p}$ estimates for the Laplace equation in perforated domains, including counterexamples and generalizations to higher dimensions.

Findings

Uniform $W^{1,p}$ estimates hold for $3/2<p<3$.

Counterexamples show estimates fail for $1<p<3/2$ and $3<p< ext{infinity}$.

Results extend to higher-dimensional domains.

Abstract

In this paper, we consider the Dirichlet problem of the three-dimensional Laplace equation in the unit ball with a shrinking hole. The problem typically arises from homogenization problems in domains perforated with tiny holes. We give an almost complete description concerning the uniform $W^{1,p}$ estimates: for any $3/2<p<3$ there hold the uniform $W^{1,p}$ estimates; for any $1<p<3/2$ or $3<p<\infty $, there are counterexamples indicating that the uniform $W^{1,p}$ estimates do not hold. The results can be generalized to higher dimensions.