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

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 $$\{\begin{array}{ll}F(D^{2}u)=f(u)& \text{in }{B}_1\cap \{u>0\},\\ u=M& \text{on }\partial B_1,\\ 0\le u<M& \text{in }{B}_1,\end{array}$$ 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.

19 pages