# Revista Matemática Iberoamericana

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**Published online first: 2019-12-04**

**DOI: 10.4171/rmi/1151**

Overdetermined problems and constant mean curvature surfaces in cones

Filomena Pacella^{[1]}and Giulio Tralli

^{[2]}(1) Università di Roma La Sapienza, Italy

(2) Università di Padova, Italy

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.

*Keywords: *Overdetermined elliptic problems, mixed boundary conditions, constant mean curvature surfaces

Pacella Filomena, Tralli Giulio: Overdetermined problems and constant mean curvature surfaces in cones. *Rev. Mat. Iberoam.* Electronically published on December 4, 2019. doi: 10.4171/rmi/1151 (to appear in print)