A Serrin-type symmetry result on model manifolds: an extension of the Weinberger argument

TL;DR

This paper extends the classical Serrin symmetry result to model manifolds with non-negative Ricci curvature by adapting Weinberger's method, establishing Euclidean symmetry under specific geometric and solution compatibility conditions.

Contribution

It introduces a novel extension of Weinberger's argument to model manifolds, linking solution symmetry to geometric properties under new compatibility assumptions.

Findings

Proves Euclidean symmetry for solutions on model manifolds with non-negative Ricci curvature.

Establishes a compatibility condition between the solution and the manifold's geometry.

Extends classical symmetry results beyond Euclidean space to curved manifolds.

Abstract

We consider the classical "Serrin symmetry result" for the overdetermined boundary value problem related to the equation $\Delta u=-1$ in a model manifold of non-negative Ricci curvature. Using an extension of the Weinberger classical argument we prove a Euclidean symmetry result under a suitable "compatibility" assumption between the solution and the geometry of the model.