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

We obtain density estimates for the free boundaries of minimizers $u\ge 0$ of the Alt-Phillips functional involving negative power potentials

$${\int }_{\mathrm{\Omega }}\left(|\mathrm{\nabla }u{|}^2+u^{-\gamma }{\chi }_{\{u>0\}}\right)dx,\gamma \in (0,2).$$

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 $\mathrm{\Gamma }$-convergence of these energies (properly rescaled) to the Dirichlet-perimeter functional

$${\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^{2}dx+Pe{r}_{\mathrm{\Omega }}(\{u=0\}),$$

considered by Athanasopoulous, Caffarelli, Kenig, and Salsa.