Overdetermined problems and constant mean curvature surfaces in cones

Filomena Pacella ^[1] ; Giulio Tralli ^[2]
1. [1] Università di Roma La Sapienza
2. [2] Università di Padova
Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 3, 2020, págs. 841-867
Idioma: inglés
DOI: 10.4171/rmi/1151
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Resumen
- 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.