The Alt-Phillips functional for negative powers

Abstract

We develop the free boundary regularity for nonnegative minimizers of the Alt-Phillips functional for negative power potentials $${\int }_{\Omega }\left(\frac{1}{2}|\nabla u{|}^2+u^{\gamma }{\chi }_{\{u>0\}}\right)dx,\gamma \in (-2,0),$$ and establish a $\Gamma$-convergence result of the rescaled energies to the perimeter functional as $\gamma \to -2$.