Finding the first nodal line of a convex domain
type: Item
instance of: scholarly article
title: Finding the first nodal line of a convex domain
zbMATH Open document ID: 0887.31002
published in: The Journal of Fourier Analysis and Applications
publication date: 1998-05-18T00:00:00Z
review text: Let \(\Omega\) be a convex open set in \(\mathbb{R}^2\). Then the Dirichlet problem has eigenvalues \(0<\lambda_1 <\lambda_2 \leq\lambda_3 \leq\dots\) Let \(u\) be the second eigenfunction: \(\Delta u= -\lambda_2u\) on \(\Omega\) and \(u=0\) on \(\partial\Omega\). The set \(\Lambda= \{z\in \Omega: u(z)=0\}\) is called the first nodal line, and it is known that \(\Omega \setminus \Lambda\) consists of two components and that \(\overline\Lambda \cap\partial \Omega \neq \emptyset\). The paper under review is concerned with the case where \(\Omega\) is long and thin. Let \((x,y)\) be Cartesian coordinates and suppose that \(\Omega\) is rotated and dilated so that its projection onto the \(y\)-axis has length 1 and this is the shortest of all projections of \(\Omega\). The main result says that \(\Lambda\) lies in a strip \([x_0-C,\;x_0+C] \times\mathbb{R}\). Here \(x_0\) is such that the domains \(\{(x,y)\in \Omega: x<x_0\}\) and \(\{(x,y)\in \Omega: x>x_0\}\) have equal first eigenvalue for the Dirichlet problem, and \(C\) is an absolute constant. The main ideas of the proof are presented in a discursive style with the aid of several examples and diagrams. The details of the proof are given in another paper of the author's [Ann. Math., II. Ser. 141, 1-33 (1995; Zbl 0831.35115)].
Mathematics Subject Classification ID: 31A25
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 49R50
zbMATH DE Number: 1115820
zbMATH Keywords: nodal line
zbMATH Keywords: convex domain
zbMATH Keywords: Dirichlet problem
author: David Jerison
reviewed by: D. H. Armitage
MaRDI profile type: Publication
Recommended article: Q4713509
Recommended article: On the nodal lines of second eigenfunctions of the fixed membrane problem
Recommended article: The nodal line of the second eigenfunction of the Laplacian in \(\mathbb{R}^2\) can be closed
Recommended article: Closed nodal surfaces for simply connected domains in higher dimensions
Recommended article: Locating the first nodal linein the Neumann problem
Recommended article: The size of the first eigenfunction of a convex planar domain
Recommended article: Locating the first nodal set in higher dimensions
Recommended article: On the lack of directional quasiconcavity of the fundamental mode in the clamped membrane problem
Recommended article: On the nodal line of the second eigenfunction of the Laplacian in \(\mathbb{R}^ 2\)
Recommended article: The nodal surface of the second eigenfunction of the Laplacian in \(\mathbb{R}^D\) can be closed
description: scientific article; zbMATH DE number 1115820