Serrin-type Overdetermined problems in $\mathbb H^n$

TL;DR

This paper proves symmetry results for overdetermined Hessian equations in hyperbolic space, extending Euclidean space results using Rellich-Pohozaev identities and P functions.

Contribution

It generalizes the symmetry results of overdetermined Hessian problems from Euclidean to hyperbolic space using novel analytical techniques.

Findings

Established symmetry of solutions in hyperbolic space

Extended Euclidean overdetermined problem results to hyperbolic setting

Developed new identities for hyperbolic Hessian equations

Abstract

In this paper, we prove the symmetry of the solution to overdetermined problem for the equation $\sigma_k(D^2u-uI)=C_n^k$ in hyperbolic space. Our approach is based on establishing a Rellich-Pohozaev type identity and using a P function. Our result generalizes the overdetermined problem for Hessian equation in Euclidean space.