Symmetry for a fully nonlinear free boundary problem with highly singular term

TL;DR

This paper proves radial symmetry of solutions to a fully nonlinear free boundary problem with a highly singular term, in both elliptic and parabolic cases, overcoming regularity challenges with novel assumptions and comparison principles.

Contribution

It establishes symmetry results for solutions with singular right-hand sides in fully nonlinear free boundary problems, extending classical methods to non-smooth contexts.

Findings

Solutions are radially symmetric in the unit ball.

The approach handles singularities where the right-hand side behaves like u^a with a in (-1,0).

The method applies to both elliptic and parabolic regimes.

Abstract

In this paper we prove radial symmetry for solutions to a free boundary problem with a singular right hand side, in both elliptic and parabolic regime. More exactly, in the unit ball $B_1$ we consider a solution to the fully nonlinear elliptic problem $$ \begin{cases} F(D^2u)=f(u)&\text{in }B_1 \cap \{u >0 \},\\ u=M&\text{on }\partial B_1,\\ 0\le u<M&\text{in }B_1,\end{cases}$$ where the right hand side $f(u) $, near $u=0$, behaves like $u^a$ with negative values for $a \in (-1,0)$. Due to lack of $C^2$-smoothness of both $u$ and the free boundary $\partial\{u>0\}$, we cannot apply the well-known Serrin-type boundary point lemma. We circumvent this by an exact assumption on a first order expansion and the decay on the second order, along with an ad-hoc comparison principle. We treat equally the parabolic case of the problem, and state a corresponding result.