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

Layan El Hajj; Seongmin Jeon; Henrik Shahgholian

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arXiv · 2207.01157

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

Layan El Hajj·Seongmin Jeon·Henrik Shahgholian

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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 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.

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Layan El Hajj, Seongmin Jeon, Henrik Shahgholian. 2022-07-04. Symmetry for a fully nonlinear free boundary problem with highly singular term. https://arxiv.org/abs/2207.01157

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