Overdetermined problems for fully nonlinear equations in space forms

Abstract

We study overdetermined problems for fully nonlinear elliptic equations in subdomains $Ø$ 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 $Ø$ 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.