On uniform estimates for Laplace equation in balls with small holes

Yong Lu

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arXiv · 1503.01103

On uniform estimates for Laplace equation in balls with small holes

Yong Lu

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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.

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Yong Lu. 2015-03-03. On uniform estimates for Laplace equation in balls with small holes. https://arxiv.org/abs/1503.01103

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