Serrin-type problem in divergence form on Riemannian manifolds

TL;DR

This paper extends classical symmetry results for Serrin-type overdetermined boundary value problems from Euclidean spaces to Riemannian manifolds with non-negative Ricci curvature, using integral identities and geometric inequalities.

Contribution

It introduces new rigidity results for divergence form problems on Riemannian manifolds, generalizing known Euclidean symmetry theorems to curved spaces.

Findings

Equality cases imply the domain is isometric to a Euclidean ball.

Derived geometric inequalities for divergence type problems on manifolds.

Extended classical symmetry results to Riemannian settings.

Abstract

In this paper, we investigate an overdetermined boundary value problem of divergence type on bounded domains in Riemannian manifolds with non-negative Ricci curvature. Using integral identities and the $P$-function method, we derive geometric inequalities and rigidity results. Under natural conditions on the nonlinearity, we prove that equality implies the domain is isometric to a Euclidean ball, thereby extending classical symmetry results to the Riemannian setting.