Serrin's type problems in warped product manifolds

TL;DR

This paper extends Serrin's overdetermined boundary value problem to warped product manifolds, establishing rigidity results using the P-function method, thus broadening the understanding of symmetry in geometric PDEs.

Contribution

It introduces Serrin's problem in the context of warped product manifolds and proves new rigidity results with the P-function approach.

Findings

Serrin's problem is solvable only under specific geometric conditions.

Rigidity results confirm symmetry of solutions in warped product settings.

The P-function method is effective in non-Euclidean geometries.

Abstract

In this paper we consider Serrin's overdetermined problems in warped product manifolds and we prove Serrin's type rigidity results by using the P-function approach introduced by Weinberger.