Quasistatic Droplet percolation

TL;DR

This paper studies the homogenization of the Hele-Shaw problem in randomly perforated domains, showing convergence of solutions and free boundaries to a homogeneous anisotropic problem as the perforation scale diminishes.

Contribution

It extends De Giorgi-Nash-Moser estimates to perforated domains and proves non-degenerate growth near free boundaries, enabling homogenization analysis.

Findings

Solutions converge uniformly to a homogenized Hele-Shaw problem

Free boundaries exhibit uniform convergence to the homogenized free boundary

Established non-degenerate growth of solutions near free boundaries

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 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.