Overdetermined problems and constant mean curvature surfaces in cones

Abstract

We consider a partially overdetermined problem in a sector-like domain Ω in a cone Σ in R N , 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.2010 Mathematics Subject Classification. 35N25, 35B06, 53A10, 53A05.

A partially overdetermined problem in a half ball

“…Proof. Our proof follows the one in Pacella-Tralli [22] closely. Instead of flattening the boundary of the barrier and using planar reflection, here we use directly spherical reflection.…”

“…In this section we will use a totally integration method to prove Theorem 1.1. Due to the lack of regularity, we need the following formula on integration by parts, see [22], lemma 2.1. (The original statement [22], lemma 2.1 is for a sector-like domain in a cone.…”

“…Due to the lack of regularity, we need the following formula on integration by parts, see [22], lemma 2.1. (The original statement [22], lemma 2.1 is for a sector-like domain in a cone. Nevertheless, the proof is applicable in our case).…”

“…A closely related overdetermined problem in a hemi-sphere has been considered by Qiu and the second author [15] (see also [7]). We also call attention to a similar partially overdetermined problem in a convex cone which has been considered recently by Pacella-Tralli [22] (see also [6] for its generalization to general elliptic operators). Their partially overdetermined BVP is of mixed Dirichlet-Neumann type.…”

Serrin’s type overdetermined problems in convex cones

“…More precisely, x 0 may coincide with O or it may be just a point of ∂Σ \ {O} and, in this case, B R (x 0 ) ∩ Σ is half a sphere lying over a flat portion of ∂Σ. Hence, it is natural to look for a characterization of symmetry in this direction, as done in [28] (see below for a more detailed description). In order to properly describe the results, we introduce some notation.…”