Rellich-Christianson type identities for the Neumann data mass of Dirichlet eigenfunctions on polytopes

2017/05/17 by Antoine Métras, Métras, Antoine
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Point processes and geometric inequalities #Spectral Theory (math.SP) #math.AP #math.SP

paper · pdf · doi:10.48550/arxiv.1705.06198

5 pages

arxiv created 2017/05/17 · openalex publication_date 2017/05/17 · arxiv updated 2017/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Dirichlet eigenvalue problem on a simple polytope. We use the Rellich identity to obtain an explicit formula expressing the Dirichlet eigenvalue in terms of the Neumann data on the faces of the polytope of the corresponding eigenfunction. The formula is particular simple for polytopes admitting an inscribed ball tangent to all the faces. Our result could be viewed as a generalization of similar identities for simplices recently found by Christianson [1][2].

Related

• Kuttler-Sigillito's Inequalities and Rellich-Christianson Identity
• Pólya's conjecture for Dirichlet eigenvalues of annuli
• Equidistribution of Neumann data mass on triangles
• Equidistribution of Neumann data mass on simplices and a simple inverse problem
• Neumann Data Mass on Perturbed Triangles