On the nodal set of the second eigenfunction of the Laplacian in symmetric domains \(\mathbb R^N \).
type: Item
instance of: scholarly article
title: On the nodal set of the second eigenfunction of the Laplacian in symmetric domains \(\mathbb R^N \).
zbMATH Open document ID: 1042.35036
author: Lucio Damascelli
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2002-12-17T00:00:00Z
review text: A previous conjecture concerning the eigenfunctions of the Laplacian in a domain \(\Omega\in\mathbb R^N \) is considered: if \(\Omega\) is connected and symmetric with respect to \(k\) orthogonal directions, then the nodal set of the eigenfunctions must intersect the boundary. The proof is given by using some symmetry properties of the eigenfunctions and the maximum principle. For \(N=2,\,k=1\) the result was proved in \textit{L. E. Payne} [Z. Angew. Math. Phys. 24, 721--729 (1973; Zbl 0272.35058)]. Some resuls concerning the periodic solutions of the Laplacian in symmetric domains are given in \textit{J. F. Bourgat} [ Rapport I.R.I.A., Roquencourt, France (1978)].
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35A30
zbMATH DE Number: 1846823
zbMATH Keywords: second eigenfunction
zbMATH Keywords: Nodal set
zbMATH Keywords: Maximum principle.
reviewed by: Gelu Paşa
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1846823