Overdetermined problems and constant mean curvature surfaces in cones

TL;DR

This paper proves rigidity results for overdetermined boundary value problems and characterizes constant mean curvature surfaces within cones, showing they are spherical caps under certain geometric conditions.

Contribution

It establishes new rigidity theorems for overdetermined problems and characterizes constant mean curvature surfaces in cones, extending classical results to cone domains.

Findings

Solutions imply the domain is a spherical sector in convex cones.

Constant mean curvature surfaces with boundary are spherical caps under convexity or radial graph conditions.

Orthogonal boundary intersection leads to characterization without additional assumptions.

Abstract

We consider a partially overdetermined problem in a sector-like domain $\Omega$ in a cone $\Sigma$ in $\mathbb{R}^N$, $N\geq 2$, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that $\Omega$ is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces $\Gamma$ with boundary which satisfy a "gluing" condition with respect to the cone $\Sigma$. We prove that if either the cone is convex or the surface is a radial graph then $\Gamma$ must be a spherical cap. Finally we show that, under the condition that the relative boundary of the domain or the surface intersects orthogonally the cone, no other assumptions are needed.