Some recent developments on the Steklov eigenvalue problem

• Published in: Revista matemática complutense Vol. 37; no. 1; pp. 1 - 161
• Main Authors: Colbois, Bruno, Girouard, Alexandre, Gordon, Carolyn, Sher, David
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.01.2024; Springer Nature B.V
• Subjects: Algebra; Analysis; Applications of Mathematics; Eigenvalues; Eigenvectors; Free boundaries; Geometry; Inverse problems; Lower bounds; Mathematics; Mathematics and Statistics; Minimal surfaces; Riemann manifold; Topology; 53A10; Inverse spectral problem; 35P05; Free boundary minimal surfaces; 35P15; Steklov spectrum; 53C21; Primary 58C40; 58J50 Secondary; Eigenvalue bounds; 35P20; 35P30
• ISSN: 1139-1138; 1988-2807
• DOI: 10.1007/s13163-023-00480-3

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.