On uniform estimates for Laplace equation in balls with small holes



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.

Mathematics Subject Classification

35J05 35B27 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Chern Institute of MathematicsNankai UniversityTianjinChina
  2. 2.Faculty of Mathematics and PhysicsMathematical Institute, Charles UniversityPrahaCzech Republic

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