Some recent developments on the Steklov eigenvalue problem

2022/12/23 by Bruno Colbois, Colbois, Bruno, Alexandre Girouard +5 · 18 citations
Mathematics · Physics and Astronomy · #35J25 #35P15 #58J50 #58J53 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2212.12528

openalex publication_date 2022/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Steklov eigenvalue problem, first introduced over 125 years ago, has seen a surge of interest in the past few decades. This article is a tour of some of the recent developments linking the Steklov eigenvalues and eigenfunctions of compact Riemannian manifolds to the geometry of the manifolds. Topics include isoperimetric-type upper and lower bounds on Steklov eigenvalues (first in the case of surfaces and then in higher dimensions), stability and instability of eigenvalues under deformations of the Riemannian metric, optimisation of eigenvalues and connections to free boundary minimal surfaces in balls, inverse problems and isospectrality, discretisation, and the geometry of eigenfunctions. We begin with background material and motivating examples for readers that are new to the subject. Throughout the tour, we frequently compare and contrast the behavior of the Steklov spectrum with that of the Laplace spectrum. We include many open problems in this rapidly expanding area.

Cited by

• Scaling inequalities for Steklov eigenvalues in space forms and sharp eigenvalue estimates on warped product manifolds
• An indefinite Coulomb interaction from the Steklov spectrum of perforated manifolds
• Upper bounds for the Steklov eigenvalues of warped products
• Payne-Philippin's overdetermined problems on compact surfaces
• The first Steklov eigenvalue bound for graphs of positive genus
• Magnetic Steklov operator on differential forms
• Reactive capacitance of flat patches of arbitrary shape
• The spectrum of Dirichlet-to-Neumann maps for radial conductivities
• Isoperimetric Bounds for Weighted Steklov Eigenvalues with Radial Weights
• Stability Estimates for Commutativity Properties of the Dirichlet-to-Neumann Operator
• Competition of small targets in planar domains: from Dirichlet to Robin and Steklov boundary condition
• Extremal Eigenvalues of Weighted Steklov Problems
• A generalized Cheeger inequality and the Steklov Problem on finite graphs
• Comparison between the first Steklov eigenvalue and algebraic connectivity on trees
• Isoperimetric inequalities involving Steklov eigenvalues on surfaces
• Second Variation Formula for Eigenvalue Functionals on Surfaces
• A note on the first Steklov eigenvalue on planar domains
• Free boundary minimal Möbius band in spherical caps

Related

• Spectral geometry of the Steklov problem
• Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space
• Spectral geometry of the Steklov problem on orbifolds
• Inequalities for the Steklov Eigenvalues
• Sharp bounds for the first eigenvalue of a fourth order Steklov problem