A Degenerate One-Phase Free Boundary Problem Arising From the Alt-Phillips Equation for Negative Powers

We study viscosity solutions for a class of degenerate one-phase free boundary problems of the form $w\mathrm{\Delta }w=h(\mathrm{\nabla }w)$. We assume the existence of a star-shaped domain $D$ such that $h<0$ in $D$, $h=0$ on $\mathrm{\partial }D$, and $h>0$ in ${\bar{D}}^c$. This class of degenerate one-phase free boundary problems arises when a canonical transformation is performed to a semilinear equation $\mathrm{\Delta }u=f(u)$, and $f$ behaves like $-\gamma u^{-(\gamma +1)}$ for some $\gamma \in (0,2)$. In this case, known as the Alt-Phillips equation for negative power potentials, $h(\rho )=c(\mathrm{\mid }\rho {\mathrm{\mid }}^2-1)$. We show existence of a viscosity solution, Lipschitz regularity, and regularity of the free boundary at flat points. Additionally, we show that as $\gamma$ converges to $2$, the free boundary converges to a minimal surface.