On uniform estimates for Laplace equation in balls with small holes

Yong Lu

Published 2015-03-03Version 1

In this paper, we consider the Dirichlet problem of the Laplace equation in the unit ball with a shrinking hole. The problem typically arises from homogenization problems in domains perforated with very small holes. In three dimensions, we show that there holds the uniform $W^{1,p}$ estimate when $3/2<p<3$. We also show that the numbers $3/2$ and $3$ are critical in the sense that for any $1<p<3/2$ or $3<p<\infty $, there are counterexamples indicating that the uniform $W^{1,p}$ estimate does not hold. The results can be generalized to higher dimensions.