Archive for Rational Mechanics and Analysis

, Volume 215, Issue 1, pp 211–281 | Cite as

Quasistatic Droplets in Randomly Perforated Domains

  • Nestor GuillenEmail author
  • Inwon Kim


We consider the Hele-Shaw problem in a randomly perforated domain with zero Neumann boundary conditions. A homogenization limit is obtained as the characteristic scale of the domain goes to zero. Specifically, we prove that the solutions as well as their free boundaries converge uniformly to those corresponding to a homogeneous and anisotropic Hele-Shaw problem set in \({\mathbb{R}^{d}}\). The main challenge when deriving the limit lies in controlling the oscillations of the free boundary. This is overcome first by extending De Giorgi–Nash–Moser type estimates to perforated domains and second by proving the almost sure non-degenerate growth of the solution near its free boundary.


Weak Solution Free Boundary Viscosity Solution Free Boundary Problem Universal Constant 
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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA

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