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Quasistatic Droplets in Randomly Perforated Domains

Abstract

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.

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Correspondence to Nestor Guillen.

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Communicated by C. Le Bris

N. Guillen is supported by NSF DMS-1201413. I. Kim is supported by NSF DMS-0970072.

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Guillen, N., Kim, I. Quasistatic Droplets in Randomly Perforated Domains. Arch Rational Mech Anal 215, 211–281 (2015). https://doi.org/10.1007/s00205-014-0777-2

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  • DOI: https://doi.org/10.1007/s00205-014-0777-2

Keywords

  • Weak Solution
  • Free Boundary
  • Viscosity Solution
  • Free Boundary Problem
  • Universal Constant