The Schiffer problem on the cylinder and on the 2-sphere

2023/03/29 by Mouhamed Moustapha Fall, Fall, Mouhamed Moustapha, Ignace Aristide Minlend +3 · 5 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2303.17036

openalex publication_date 2023/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the existence of a family of compact subdomains Ω of the flat cylinder ℝ^N× ℝ/2πℤ for which the Neumann eigenvalue problem for the Laplacian on Ω admits eigenfunctions with constant Dirichlet values on ∂ Ω. These domains Ω have the property that their boundaries ∂ Ω have nonconstant principal curvatures. In the context of ambient Riemannian manifolds, our construction provides the first examples of such domains whose boundaries are neither homogeneous nor isoparametric hypersurfaces. The functional analytic approach we develop in this paper overcomes an inherent loss of regularity of the problem in standard function spaces. With the help of this approach, we also construct a related family of subdomains of the 2-sphere S². By this we disprove a conjecture in \citeSouam.

Cited by

• On the Schiffer and Berenstein conjectures for centrally symmetric convex domains in the plane
• A contractible Schiffer counterexample on the half-sphere
• Strong stability and the Schiffer Conjecture for the fluid-elastic semigroup
• New Homogeneous Solutions for the One-Phase Free Boundary Problem
• Non-symmetric solutions to an overdetermined problem for the Helmholtz equation in the plane

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