The Alt-Phillips functional for negative powers

TL;DR

This paper studies the regularity of solutions to a free boundary problem involving the Alt-Phillips functional with negative power potentials and shows that rescaled energies converge to the perimeter functional as the power approaches -2.

Contribution

It introduces new regularity results for minimizers of the Alt-Phillips functional with negative powers and proves a Gamma-convergence to the perimeter functional as the power approaches -2.

Findings

Established free boundary regularity for nonnegative minimizers.

Proved Gamma-convergence of rescaled energies to perimeter functional.

Analyzed the behavior of the functional as the negative power approaches -2.

Abstract

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