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

2022/07/04 by Layan El Hajj, Seongmin Jeon, Hajj, Layan El +3
Computer Science · Mathematics · #35B06 #35R35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2207.01157

openalex publication_date 2022/07/04 · openalex created_date 2022/07/07 · openalex updated_date 2026/07/28

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 B₁ we consider a solution to the fully nonlinear elliptic problem \begincases F(D²u)=f(u)&in B₁ ∩ \u >0 \, u=M&on ∂ B₁, 0≤ u0\, 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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