Some Results on the Schiffer's Conjecture in R²

2011/11/30 by Jian Deng, Deng, Jian · 4 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics #math.AP #math.DG

paper · pdf · doi:10.48550/arxiv.1112.0207

12 pages, third version, accepted by JDE

openalex publication_date 2011/11/30 · arxiv created 2012/05/18 · arxiv updated 2012/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ω be an open, bounded domain in the plane with connected and smooth boundary, and ω an eigenfunction of the Neumann Laplacian corresponding to some Neumann eigenvalue μ> 0. If the boundary value of ω is a nonzero constant along the boundary, denoting 0 = μ₁(Ω) < μ₂(Ω) <= ... the set of all Neumann eigenvalues for the Laplacian on Ω, we show that 1) if μ< μ₈(Ω); or 2) if Ω is strictly convex and centrally symmetric, μ< μ₁3(Ω), then Ω must be a disk.

Cited by

• On the Schiffer and Berenstein conjectures for centrally symmetric convex domains in the plane
• A contractible Schiffer counterexample on the half-sphere
• Non-symmetric solutions to an overdetermined problem for the Helmholtz equation in the plane
• A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures

Related

• The Schiffer problem on the cylinder and on the 2-sphere
• On the Schiffer and Berenstein conjectures for centrally symmetric convex domains in the plane
• Inequalities between Neumann and Dirichlet Laplacian eigenvalues on planar domains
• A contractible Schiffer counterexample on the half-sphere
• A short note on the Schiffer's conjecture for a class of centrally symmetric convex domains in ℝ²