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 


  1. 1.
    Allaire, G.: Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes. Arch. Ration. Mech. Anal. 113(3), 209–259 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Allaire, G.: Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. II. Noncritical sizes of the holes for a volume distribution and a surface distribution of holes. Arch. Ration. Mech. Anal. 113(3), 261–298 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brown, R.M., Shen, Z.: Estimates for the Stokes operator in Lipschitz domains. Indiana Univ. Math. J. 44(4), 1183–1206 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Caffarelli, L.A., Peral, I.: On \(W^{1, p}\) estimates for elliptic equations in divergence form. Commun. Pure Appl. Math. 51(1), 1–21 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Feireisl, E., Lu, Y.: Homogenization of stationary Navier-Stokes equations in domains with tiny holes. J. Math. Fluid Mech. 17, 381–392 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Feireisl, E., Novotný, A., Takahashi, T.: Homogenization and singular limits for the complete Navier-Stokes-Fourier system. J. Math. Pures Appl. 94(1), 33–57 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jerison, D., Kenig, C.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130(1), 161–219 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    W. Jäger, A. Mikelić. Homogenization of the Laplace equation in a partially perforated domain. In memory of Serguei Kozlov, volume 50 of Advances in Mathematics for Applied Sciences, pp 259–284 (1999)Google Scholar
  9. 9.
    Kozono, H., Sohr, H.: New a priori estimates for the Stokes equations in exterior domains. Indiana Univ. Math. J. 40(1), 1–27 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Masmoudi, N.: Homogenization of the compressible Navier–Stokes equations in a porous medium. ESAIM Control Optim. Calc. Var. 8, 885–906 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mikelić, A.: Homogenization of nonstationary Navier–Stokes equations in a domain with a grained boundary. Ann. Mat. Pura Appl. 158, 167–179 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

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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