Delaunay type domains for an overdetermined elliptic problem in Sⁿ x R\n and Hⁿ x R

2013/05/28 by Filippo Morabito, Pieralberto Sicbaldi, Morabito, Filippo +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1305.6516

openalex publication_date 2013/05/28 · openalex created_date 2022/09/04 · openalex updated_date 2026/07/28

Abstract

We prove the existence of a countable family of Delaunay type domains\n\Ω_j in Mⁿ x R, where Mⁿ is the Riemannian manifold Sⁿ or Hⁿ and n is\nat least 2, bifurcating from the cylinder Bⁿ x R (where Bⁿ is a geodesic ball\nof radius 1 in Mⁿ) for which the first eigenfunction of the Laplace-Beltrami\noperator with zero Dirichlet boundary condition also has constant Neumann data\nat the boundary. The domains \Ω_j are rotationally symmetric and periodic\nwith respect to the R-axis of the cylinder and as j converges to 0 the domain\n\Ω_j converges to the cylinder Bⁿ x R.\n

Related

• Bifurcating extremal domains for the first eigenvalue of the Laplacian
• Overdetermined elliptic problems in onduloid-type domains with general nonlinearities
• Overdetermined problems with sign-changing eigenfunctions in unbounded periodic domains
• Bifurcating domains for an overdetermined eigenvalue problem in cylinders
• Sign-changing solution for an overdetermined elliptic problem on unbounded domain