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

Abstract

In this paper, we prove the symmetry of the solution to overdetermined problem for the equation ${\sigma }_k(D^{2}u-uI)=C_n^k$ 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.