A partially overdetermined problem in a half ball

Abstract

In this paper, we study a partially overdetermined mixed boundary value problem in a half ball. We prove that a domain in which this partially overdetermined problem admits a solution if and only if the domain is a spherical cap intersecting S n−1 orthogonally. As an application, we show a stationary point for a partially torsional rigidity under a volume constraint must be a spherical cap.

Isoperimetric cones and minimal solutions of partial overdetermined problems

“…Let us observe that the hypothesis that the solution u belongs to W 1,∞ (Ω) ∩ W 2,2 (Ω) is automatically satisfied when Γ and Γ 1 intersect orthogonally, as proved in [18]. We also refer the reader to the recent works [6,9,13].…”

Radial symmetry of solutions to anisotropic and weighted diffusion equations with discontinuous nonlinearities

“…We remark that the main results in [L], focusing on elliptic semilinear equations in the plane, can be seen as a counterpart of those in [GNN], in the sense that the results in [L] weaken the smoothness assumptions on the source term and on the solution with respect to the setting in [GNN], at the expense of restricting to positive nonlinearities and to dimension two only. Interestingly, the method in [L], being based on integral (in)equalities, is conceptually different from the moving plane technique used in [GNN] and, in a sense, it is more related, apart from several important structural differences, to the approach to radial symmetry inhaugurated by Weinberger in [W]: see also, e.g., [PSc,Re,GL,FGK,FK,BNST,MP,MP2,MP3,GS,EP,Po,CS,WX,BC,GX,DPV,PT,PT2,CG,CiR] for related problems and ramifications.…”

A partially overdetermined problem in domains with partial umbilical boundary in space forms

“…In particular, Alexandrov type theorem says that a free boundary CMC hypersurface in a half ball must be a free boundary spherical cap. Motivated by this, we have proposed in [10] the study of a partially overdetemined BVP in a half ball. Precisely, let B n + = {x ∈ B n : x n > 0} be the half Euclidean unit ball and Ω ⊂ B n…”

“…Theorem 1.1. ( [10]) Let Ω be as above. Assume (1.2) admits a weak solution u ∈ W 1,2 0 (Ω, Σ) = {u ∈ W 1,2 (Ω), u|Σ = 0}, i.e., Ω ( ∇u, ∇v + v) dx − T uv dA = 0, for all v ∈ W 1,2 0 (Ω, Σ).…”

“…Since the Euclidean case K = 0 has already been handled in [10], and the case κ = 0 in M n (K) has been considered in [5] (as a special case of a flat cone), in this paper we consider the hyperbolic case K = −1 with κ > 0 and the spherical case K = 1 with κ > 0.…”