Radial symmetry and partially overdetermined problems in a convex cone

TL;DR

This paper proves radial symmetry and establishes new results for overdetermined boundary value problems in convex cones, using maximum principles, integral identities, and eigenvalue analysis in space forms.

Contribution

It introduces novel symmetry results and identities for overdetermined problems in convex cones, extending classical results to new geometric settings.

Findings

Radial symmetry of solutions in convex cones.

Serrin-type results for problems outside convex cones in dimension 2.

A Rellich identity for eigenvalue problems with mixed boundary conditions.

Abstract

We obtain the radial symmetry of the solution to a partially overdetermined boundary value problem in a convex cone in space forms by using the maximum principle for a suitable subharmonic function $P$ and integral identities. In dimension $2$, we prove Serrin-type results for partially overdetermined problems outside a convex cone. Furthermore, we obtain a Rellich identity for an eigenvalue problem with mixed boundary conditions in a cone.