Uniform density estimates and $ \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

\begin{document}$ \int_\Omega \left(|\nabla u|^2 + u^{-\gamma} \chi_{\{u>0\}}\right) \, dx, \quad \quad \gamma \in (0, 2). $\end{document}

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

\begin{document}$ \int_{\Omega} |\nabla u|^2 dx + Per_{\Omega}(\{ u = 0\}), $\end{document}

considered by Athanasopoulous, Caffarelli, Kenig, and Salsa.