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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 (3/2u003cpu003c3) there hold the uniform (W^{1,p}) estimates; for any (1u003cpu003c3/2) or (3u003cpu003cinfty ), there are counterexamples indicating that the uniform (W^{1,p}) estimates do not hold. The results can be generalized to higher dimensions.

On uniform estimates for Laplace equation in balls with small holes