Overdetermined problems for fully nonlinear equations in space forms

TL;DR

This paper proves that for certain fully nonlinear elliptic equations in space forms, the only domains supporting solutions are geodesic balls, extending known results from semilinear cases to fully nonlinear equations.

Contribution

It establishes a new overdetermined problem result for fully nonlinear equations in space forms, generalizing previous semilinear findings.

Findings

Solutions exist only in geodesic balls

Extends classical overdetermined results to fully nonlinear equations

Applicable in Euclidean sphere and hyperbolic space

Abstract

We study overdetermined problems for fully nonlinear elliptic equations in subdomains $\O$ of the Euclidean sphere $\mathbb{S}^{N}$ and the hyperbolic space $\mathbb{H}^{N}$. We prove, the existence of a classical solution to the underlined equation forces $\O$ to be a geodesic ball in the ambient space. Our result extends to fully nonlinear equations, a similar result in the case of semilinear equations with the Laplace operator due to Kumaresan and Prajapat.