Rigidity results for Serrin's overdetermined problems in Riemannian manifolds

TL;DR

This paper establishes rigidity and symmetry results for Serrin's overdetermined problems on Riemannian manifolds with conformal vector fields, extending classical Euclidean results to a geometric setting.

Contribution

It introduces a Pohozaev-type identity in Riemannian manifolds and demonstrates rigidity and symmetry results using conformal changes and geometric identities.

Findings

Proves a Pohozaev-type identity for manifolds with conformal vector fields.

Establishes a Serrin-type rigidity result in Riemannian manifolds.

Derives a symmetry result for the Dirichlet problem using a generalized shear stress bound.

Abstract

In this work, we are interested in studying Serrin's overdetermined problems in Riemannian manifolds. For manifolds endowed with a conformal vector field, we prove a Pohozoaev-type identity to show a Serrin's type rigidity result using the P-function approach introduced by Weinberger. We proceed with a conformal change to achieve this goal, starting from a geometric Pohozaev identity due to Schoen. Moreover, we obtain a symmetry result for the associated Dirichlet problem by using a generalized normalized wall shear stress bound.