Spectral geometry of the Steklov problem

Abstract

The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometry. While this problem shares some common properties with its more familiar Dirichlet and Neumann cousins, its eigenvalues and eigenfunctions have a number of distinctive geometric features, which makes the subject especially appealing. In this survey we discuss some recent advances and open questions, particularly in the study of spectral asymptotics, spectral invariants, eigenvalue estimates, and nodal geometry.

arXiv comment26 pages, 7 figures

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• Distribution of first-reaction times with target regions on boundaries of shell-like domains
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• Stability from rigidity via umbilicity
• Geometry and interior nodal sets of Steklov eigenfunctions
• Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map
• Escobar's Conjecture on a sharp lower bound for the first nonzero Steklov eigenvalue
• Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on its boundary in Riemannian manifolds
• A higher dimensional generalization of Hersch-Payne-Schiffer inequality for Steklov eigenvalues
• Spectral asymptotics for Dirichlet to Neumann operator
• Boundary Integrals and Approximations of Harmonic Functions
• Monotonicity results for the first Steklov eigenvalue on compact surfaces
• Steklov Spectral Geometry for Extrinsic Shape Analysis
• Tunneling for the Robin Laplacian in smooth planar domains
• Interior nodal sets of Steklov eigenfunctions on surfaces
• Compact manifolds with fixed boundary and large Steklov eigenvalues
• Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d -sets
• A Numerical Study of Steklov Eigenvalue Problem via Conformal Mapping
• Hypersurfaces of Euclidean space with prescribed boundary and small Steklov eigenvalues
• Trace and inverse trace of Steklov eigenvalues II
• Hausdorff measure bounds for nodal sets of Steklov eigenfunctions
• Laplacian Eigenproblems on Product Regions and Tensor Products of Sobolev Spaces