Uniform density estimates and $\Gamma$-convergence for the Alt-Phillips functional of negative powers

TL;DR

This paper studies the behavior of minimizers of a free boundary problem involving negative powers, establishing uniform density estimates and convergence of free boundaries to minimal surfaces as the parameter approaches a critical value.

Contribution

It provides uniform density estimates and proves $ ext{Gamma}$-convergence of the Alt-Phillips functional to a Dirichlet-perimeter functional as the parameter approaches 2.

Findings

Uniform density estimates for free boundaries as $ ext{gamma} o 2$

Convergence of free boundaries to minimal surfaces

Gamma-convergence of energies to Dirichlet-perimeter functional

Abstract

We obtain density estimates for the free boundaries of minimizers $u \ge 0$ of the Alt-Phillips functional involving negative power potentials $$\int_\Omega \left(|\nabla u|^2 + u^{-\gamma} \chi_{\{u>0\}}\right) \, dx, \quad \quad \gamma \in (0,2).$$ These estimates remain uniform as the parameter $\gamma \to 2$. As a consequence we establish the uniform convergence of the corresponding free boundaries to a minimal surface as $\gamma \to 2$. The results are based on the $\Gamma$-convergence of these energies (properly rescaled) to the Dirichlet-perimeter functional $$\int_\Omega |\nabla u|^2 dx + Per_{\Omega}(\{ u=0\}),$$ considered by Athanasopoulous, Caffarelli, Kenig, and Salsa.