Nearly radial Neumann eigenfunctions on symmetric domains
type: Item
instance of: scholarly article
title: Nearly radial Neumann eigenfunctions on symmetric domains
zbMATH Open document ID: 1396.35044
DOI: 10.4171/JST/218
published in: Journal of Spectral Theory
publication date: 2018-08-31T00:00:00Z
review text: Summary: We study the existence of Neumann eigenfunctions of the Laplacian which do not change sign on the boundary of some special domains. We show that eigenfunctions which are strictly positive on the boundary exist on regular polygons with at least 6 sides, while on equilateral triangles and cubes it is not even possible to find an eigenfunction which is nonnegative on the boundary. This work builds on recent results of Hoffmann-Ostenhof about rectangles.{ }We use analytic methods combined with symmetry arguments to prove the result for polygons with six or more sides, and combinatorics for equilateral triangles and cubes. The case of the regular pentagon is much harder. Its proof requires deep computational and numerical results which are beyond the scope of the present paper. The pentagonal case, codeveloped with Nilima Nigam and Benjamin Young, will appear in a companion paper.
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 6930089
zbMATH Keywords: eigenvalue bounds
zbMATH Keywords: nodal line
zbMATH Keywords: triangles
author: Bartłomiej Andrzej Siudeja
MaRDI profile type: Publication
Wikidata QID: Q129524087
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description: scientific article; zbMATH DE number 6930089