Eigenvalue and gap estimates of isometric immersions for the Dirichlet-to-Neumann operator acting on \(p\)-forms
type: Item
instance of: scholarly article
title: Eigenvalue and gap estimates of isometric immersions for the Dirichlet-to-Neumann operator acting on \(p\)-forms
zbMATH Open document ID: 1410.58014
author: Deborah Michel
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2019-03-12T00:00:00Z
review text: Let \((M,g)\) be an \((n+1)\)-dimensional compact oriented Riemannian manifold with smooth boundary \(\partial M\) isometrically immersed in some Euclidean space. By using the extension of the Dirichlet-to-Neumann operator (also called Steklov operator) \(T^{[p]}\) acting on the bundle of \(p\)-forms \(\bigwedge^p(\partial M)\) developed in [\textit{S. Raulot} and \textit{A. Savo}, J. Funct. Anal. 262, No. 3, 889--914 (2012; Zbl 1250.58013)], the article under review gives a lower bound for \(\nu_{1,p}-\nu_{1,p-1}\), where \(\nu_{k,p}\) denotes the \(k\)-th eigenvalue of \(T^{[p]}\), in terms of the curvature of \(M\). The proof follows the method developed in [\textit{P. Guerini} and \textit{A. Savo}, Trans. Am. Math. Soc. 356, No. 1, 319--344 (2004; Zbl 1034.58026)], where the Hodge-Laplace operator acting on \(p\)-forms was considered.
reviewed by: Emilio A. Lauret
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 58J32
Mathematics Subject Classification ID: 35P15
zbMATH DE Number: 7034594
zbMATH Keywords: Dirichlet-to-Neumann operator
zbMATH Keywords: Steklov operator
zbMATH Keywords: \(p\)-forms
zbMATH Keywords: first eigenvalue
zbMATH Keywords: Riemannian manifold with boundary
zbMATH Keywords: differential forms
MaRDI profile type: Publication
full work available at URL: j.crma.2019.01.006
OpenAlex ID: W2911857009
cites work: An isoperimetric inequality and the first Steklov eigenvalue
cites work: A comparison theorem for the first non-zero Steklov eigenvalue
cites work: Opérateur de courbure et laplacien des formes différentielles d'une variété riemannienne
cites work: Eigenvalue and gap estimates for the Laplacian acting on 𝑝-forms
cites work: A Reilly formula and eigenvalue estimates for differential forms
cites work: On the first eigenvalue of the Dirichlet-to-Neumann operator on forms
cites work: On the spectrum of the Dirichlet-to-Neumann operator acting on forms of a Euclidean domain
cites work: Hodge decomposition. A method for solving boundary value problems
Wikidata QID: Q128417102
DOI: 10.1016/J.CRMA.2019.01.006
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description: scientific article; zbMATH DE number 7034594