Uniform density estimates and $Γ$-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 $\int_Ω\left(|\nabla u|^2 + u^{-γ} χ_{\{u>0\}}\right) \, dx, \quad \quad γ\in (0,2).$ These estimates remain uniform as the parameter $γ\to 2$. As a consequence we establish the uniform convergence of the corresponding free boundaries to a minimal surface as $γ\to 2$. The results are based on the $Γ$-convergence of these energies (properly rescaled) to the Dirichlet-perimeter functional $\int_Ω|\nabla u|^2 dx + Per_Ω(\{ u=0\}),$ considered by Athanasopoulous, Caffarelli, Kenig, and Salsa.

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