Serrin's type overdetermined problems in convex cones

TL;DR

This paper investigates Serrin's overdetermined boundary value problems within convex cones, demonstrating that solutions imply the domain must be a spherical sector, extending results to degenerate operators and space forms.

Contribution

It generalizes Serrin's overdetermined problems to convex cones and space forms, establishing rigidity results for degenerate operators.

Findings

Solutions imply the domain is a spherical sector

Extends rigidity results to degenerate operators

Includes generalization to space forms

Abstract

We consider overdetermined problems of Serrin's type in convex cones for (possibly) degenerate operators in the Euclidean space as well as for a suitable generalization to space forms. We prove rigidity results by showing that the existence of a solution implies that the domain is a spherical sector.