Publication at Faculty of Mathematics and Physics |
2016
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,W- p estimates: for any 3/2 < p < 3 there hold the uniform W-1,W- p estimates; for any 1 < p < 3/ 2 or 3 < p < 8, there are counterexamples indicating that the uniform W-1,W- p estimates do not hold. The results can be generalized to higher dimensions.