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

Abstract

We obtain density estimates for the free boundaries of minimizers $u\ge 0$ of the Alt-Phillips functional involving negative power potentials $${\int }_{\Omega }\left(|\nabla u{|}^2+u^{-\gamma }{\chi }_{\{u>0\}}\right)dx,\gamma \in (0,2).$$ 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 $${\int }_{\Omega }|\nabla u{|}^{2}dx+Pe{r}_{\Omega }(\{u=0\}),$$ considered by Athanasopoulous, Caffarelli, Kenig, and Salsa.