$C^\infty$ regularity in semilinear free boundary problems

TL;DR

This paper proves that solutions and free boundaries in a class of semilinear free boundary problems become infinitely smooth once they reach a certain regularity, using advanced estimates for equations with boundary-singular potentials.

Contribution

It establishes $C^ abla$ regularity for solutions and free boundaries in the Alt-Phillips problem, extending to $C^ abla$ smoothness and detailed regularity estimates for boundary-singular equations.

Findings

Free boundaries are $C^ abla$ once $C^{1, abla}$

Solutions are $C^ abla$ after boundary regularity

Includes critical Hardy potential case $rac{1}{4}$

Abstract

We study the higher regularity of solutions and free boundaries in the Alt-Phillips problem $\Delta u=u^{\gamma-1}$, with $\gamma\in(0,1)$. Our main results imply that, once free boundaries are $C^{1,\alpha}$, then they are $C^\infty$. In addition $u/d^{\frac{2}{2-\gamma}}$ and $u^{\frac{2-\gamma}{2}}$ are $C^\infty$ too. In order to achieve this, we need to establish fine regularity estimates for solutions of linear equations with boundary-singular Hardy potentials $-\Delta v = \kappa v/d^2$ in $\Omega$, where $d$ is the distance to the boundary and $\kappa\leq\frac{1}{4}$. Interestingly, we need to include even the critical constant $\kappa=\frac{1}{4}$, which corresponds to $\gamma=\frac{2}{3}$.