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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Communicated by F. H. Lin.
The author thanks E. Feireisl, C. Prange, S. Schwarzacher and J. Žabenský for interesting discussions. The author acknowledges the support of the project LL1202 in the program ERC-CZ funded by the Ministry of Education, Youth and Sports of the Czech Republic.
