Some recent developments on the Steklov eigenvalue problem

B Colbois, A Girouard, C Gordon, D Sher - Revista Matemática …, 2024 - Springer
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 …

The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander's rediscovered manuscript

A Girouard, M Karpukhin, M Levitin… - Journal of Spectral …, 2022 - ems.press
How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding
boundary Laplacian? This question has been actively investigated in recent years …

A numerical study of the generalized Steklov problem in planar domains

A Chaigneau, DS Grebenkov - Journal of Physics A …, 2024 - iopscience.iop.org
We numerically investigate the generalized Steklov problem for the modified Helmholtz
equation and focus on the relation between its spectrum and the geometric structure of the …

Robust superlinear Krylov convergence for complex noncoercive compact-equivalent operator preconditioners

O Axelsson, J Karátson, F Magoulès - SIAM Journal on Numerical Analysis, 2023 - SIAM
Preconditioning for Krylov methods often relies on operator theory when mesh independent
estimates are looked for. The goal of this paper is to contribute to the long development of …

A numerical study of the Dirichlet-to-Neumann operator in planar domains

A Chaigneau, DS Grebenkov - arxiv preprint arxiv:2310.19571, 2023 - arxiv.org
We numerically investigate the generalized Steklov problem for the modified Helmholtz
equation and focus on the relation between its spectrum and the geometric structure of the …

The Steklov problem for exterior domains: asymptotic behavior and applications

DS Grebenkov, A Chaigneau - arxiv preprint arxiv:2407.09864, 2024 - arxiv.org
We investigate the spectral properties of the Steklov problem for the modified Helmholtz
equation $(p-\Delta) u= 0$ in the exterior of a compact set, for which the positive parameter …

[HTML][HTML] Applications of possibly hidden symmetry to Steklov and mixed Steklov problems on surfaces

T Arias-Marco, EB Dryden, CS Gordon… - Journal of Mathematical …, 2024 - Elsevier
We consider three different questions related to the Steklov and mixed Steklov problems on
surfaces. These questions are connected by the techniques that we use to study them, which …

Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues

M Levitin, L Parnovski, I Polterovich… - Journal d'Analyse …, 2022 - Springer
In the present paper we develop an approach to obtain sharp spectral asymptotics for
Steklov type problems on planar domains with corners. Our main focus is on the two …

Sloshing, Steklov and corners: asymptotics of sloshing eigenvalues

M Levitin, L Parnovski, I Polterovich… - arxiv preprint arxiv …, 2017 - arxiv.org
In the present paper we develop an approach to obtain sharp spectral asymptotics for
Steklov type problems on planar domains with corners. Our main focus is on the two …

Mixed Steklov-Neumann problem: asymptotic analysis and applications to diffusion-controlled reactions

DS Grebenkov - arxiv preprint arxiv:2409.00213, 2024 - arxiv.org
Many first-passage processes in complex media and related diffusion-controlled reactions
can be described by means of eigenfunctions of the mixed Steklov-Neumann problem. In …