We consider a partially overdetermined problem in a sector-like domain Ω in a cone Σ in RN, N ≥ 2, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that Ω is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces Γ with boundary which satisfy a ‘gluing’ condition with respect to the cone Σ. We prove that if either the cone is convex or the surface is a radial graph then Γ 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.

Overdetermined problems and constant mean curvature surfaces in cones

Tralli G.
2020

Abstract

We consider a partially overdetermined problem in a sector-like domain Ω in a cone Σ in RN, N ≥ 2, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that Ω is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces Γ with boundary which satisfy a ‘gluing’ condition with respect to the cone Σ. We prove that if either the cone is convex or the surface is a radial graph then Γ 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/3351409
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