Eigenvalues estimate for the Neumann problem of a bounded domain
type: Item
instance of: scholarly article
title: Eigenvalues estimate for the Neumann problem of a bounded domain
zbMATH Open document ID: 1158.58014
author: Bruno Colbois
author: Daniel Maerten
published in: The Journal of Geometric Analysis
publication date: 2008-10-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/0802.2774
review text: Let \((M,g)\) be a complete \(n\)-dimensional Riemannian manifold with Ricci curvature bounded below by \(-(n-1)a^2\) for some \(a\geq0\). Let \(\Omega\) be a bounded domain in \((M,g)\) with smooth boundary of volume \(V(\Omega)\). Denote the eigenvalues of the Neumann Laplacian on \(\Omega\) by \(0=\lambda_1(\Omega)\leq\dots \leq\Omega_k(\Omega)\leq\dots\). The authors show: Theorem: There exist constants \(A_n>0\) and \(B_n>0\) so that \(\lambda_k(\Omega)\leq A_na^2+B_n({k\over V(\Omega)})^{2/n}\).
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 53C99
Mathematics Subject Classification ID: 35P15
zbMATH DE Number: 5357888
zbMATH Keywords: Neumann spectrum
zbMATH Keywords: upper bound
zbMATH Keywords: Weyl law
zbMATH Keywords: metric geometry
zbMATH Keywords: Buser upper bound
zbMATH Keywords: Ricci curvature
zbMATH Keywords: Rayleigh quotient
reviewed by: Peter B. Gilkey
MaRDI profile type: Publication
OpenAlex ID: W2009796115
arXiv ID: 0802.2774
cites work: Q3806386
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cites work: Q3134551
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cites work: Upper bounds for eigenvalues of conformal metrics
cites work: Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space
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DOI: 10.1007/S12220-008-9041-Z
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description: scientific article