Symmetry and rigidity results for Serrin's overdetermined type problems in weighted Riemannian manifolds

TL;DR

This paper investigates symmetry and rigidity in Serrin's overdetermined problems within weighted Riemannian manifolds, establishing geometric characterizations under various domain conditions.

Contribution

It provides new rigidity results for solutions and geometry in weighted Riemannian manifolds, including non-compact and singular cases with radial weights.

Findings

Rigidity results for compact domains in weighted Riemannian manifolds.

Characterization of solutions and geometry in conformally Euclidean spaces with radial weights.

Analysis of domains with non-compact closure and potential singularities.

Abstract

We study Serrin's overdetermined boundary value problems in bounded domains on weighted Riemannian manifolds. When the closure of the domain is compact, we establish a rigidity result that characterizes both the solution and the geometry of the ambient manifold. We further address the case of domains with non-compact closure for manifolds conformally equivalent to the Euclidean space, possibly degenerating or becoming singular at a point, where both the weight and the conformal factor are radial functions.