Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on its boundary in Riemannian manifolds

2017/04/07 by Changwei Xiong, Xiong, Changwei · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical methods in inverse problems #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1704.02073

openalex publication_date 2017/04/07 · openalex created_date 2017/04/28 · openalex updated_date 2026/07/28

Abstract

We prove that in Riemannian manifolds the k-th Steklov eigenvalue on a domain and the square root of the k-th Laplacian eigenvalue on its boundary can be mutually controlled in terms of the maximum principal curvature of the boundary under sectional curvature conditions. As an application, we derive a Weyl-type upper bound for Steklov eigenvalues. A Pohozaev-type identity for harmonic functions on the domain and the min-max variational characterization of both eigenvalues are important ingredients.

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• A weighted Reilly type integral formula for differential forms and its applications
• Payne-Philippin's overdetermined problems on compact surfaces
• Compact manifolds with fixed boundary and large Steklov eigenvalues

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