  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 a 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 surely non-degenerate growth of the solution near its free boundary.

Analysis of PDEs