Nodal set openings on perturbed rectangular domains
type: Item
instance of: scholarly article
title: Nodal set openings on perturbed rectangular domains
author: Thomas L. Beck
author: Marichi Gupta
author: Jeremy L. Marzuola
published in: Annales Henri Poincaré
publication date: 2024-10-31T00:00:00Z
review text: The authors investigate how the nodal set of eigenfunctions of the Dirichlet Laplacian changes with respect to a perturbation of domain's boundary. More precisely, let \(\mathcal{R} = (0,N) \times (0,1)\) be a planar rectangle such that the corresponding eigenvalue \(\lambda_{2,2}\) is simple. Let \(\Omega\) be the perturbation of \(\mathcal{R}\) defined as \((\eta \phi_L(\cdot),N) \times (0,1)\), where \(\eta>0\) is sufficiently small and the function \(\phi_L\) is sufficiently regular, \(-1 \leq \phi_L \leq 0\), and \N\[\N\int_0^1 \phi_L(y) \sin(2\pi y) \sin(\pi y) \,dy \neq 0. \N\]\NUnder these assumptions, the authors study an eigenfunction \(v\) in \(\Omega\) which is a perturbation of the eigenfunction \(\varphi_{2,2}\) in \(\mathcal{R}\). In the main results, it is shown that the nodal set of \(v\) develops (with respect to \(\eta\)) an opening near the point of crossing of nodal lines of \(\varphi_{2,2}\), and this opening is analyzed both qualitatively and quantitatively. In particular, the orientation, size, shape, and regularity of the nodal set of \(v\) are investigated. Apart from many other tools, the authors carefully work with the Fourier expansion of \(v\). Numerical simulations are also given.
reviewed by: Vladimir Bobkov
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35B20
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 7936712
zbMATH Keywords: eigenfunctions
zbMATH Keywords: nodal set
zbMATH Keywords: crossing of nodal lines
zbMATH Keywords: perturbation
zbMATH Keywords: approximate separation of variables
MaRDI profile type: Publication
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DOI: 10.1007/S00023-024-01424-3
description: scientific article; zbMATH DE number 7936712