# Overdetermined problems and constant mean curvature surfaces in cones

### Filomena Pacella

Università di Roma La Sapienza, Italy### Giulio Tralli

Università di Padova, Italy

## 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.

## Cite this article

Filomena Pacella, Giulio Tralli, Overdetermined problems and constant mean curvature surfaces in cones. Rev. Mat. Iberoam. 36 (2020), no. 3, pp. 841–867

DOI 10.4171/RMI/1151