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

TL;DR

This paper numerically explores the spectral properties of the Dirichlet-to-Neumann operator in planar domains, revealing how eigenvalues and eigenfunctions relate to domain geometry and symmetry, with implications for spectral geometry and diffusion processes.

Contribution

It provides new numerical insights into the spectrum of the Dirichlet-to-Neumann operator for various planar domains, including asymptotic behavior and eigenfunction localization.

Findings

Eigenvalues exhibit specific asymptotic patterns in polygonal domains

Eigenfunctions' integrals depend on domain symmetries

Eigenfunctions decay exponentially away from the boundary in smooth shapes

Abstract

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 domain. We address three distinct aspects: (i) the asymptotic behavior of eigenvalues for polygonal domains; (ii) the dependence of the integrals of eigenfunctions on the domain symmetries; and (iii) the localization and exponential decay of Steklov eigenfunctions away from the boundary for smooth shapes and in the presence of corners. For this purpose, we implemented two complementary numerical methods to compute the eigenvalues and eigenfunctions of the associated Dirichlet-to-Neumann operator for various simply-connected planar domains. We also discuss applications of the obtained results in the theory of diffusion-controlled reactions and formulate several conjectures with relevance in spectral geometry.