On the fully nonlinear Alt-Phillips equation

TL;DR

This paper investigates the fully nonlinear Alt-Phillips equation for parameters between 1 and 2, establishing optimal regularity of solutions and smoothness of the free boundary.

Contribution

It provides the first regularity results for solutions and free boundaries in the fully nonlinear Alt-Phillips equation for the specified parameter range.

Findings

Solutions have optimal regularity.

The free boundary's regular part is $C^1$.

Results extend understanding of nonlinear free boundary problems.

Abstract

For a parameter $\gamma\in(1,2)$, we study the fully nonlinear version of the Alt-Phillips equation, $F(D^2u)=u^{\gamma-1}$, for $u\ge 0.$ We establish the optimal regularity of the solution, as well as the $C^1$ regularity of the regular part of the free boundary.