On conditions for transmission in the Sturm-Liouville problem on a network
type: Item
instance of: scholarly article
title: On conditions for transmission in the Sturm-Liouville problem on a network
zbMATH Open document ID: 1098.34021
author: Yu. V. Pokornij
author: V. L. Pryadiev
published in: Journal of Mathematical Sciences (New York)
publication date: 2006-01-23T00:00:00Z
review text: The authors study various approaches to transmission problems on graphs, on the basis of their reduction to problems for a single differential equation on the graph. For example, just as vibrations of a string can be modelled by a Sturm-Liouville problem on a line segment, vibrations of a woven membrane can be modelled by a Sturm-Liouville problem on a metric graph. The authors show that, for all \(M\) and all sufficiently small \(h\), the natural frequencies less than \(M\) of a uniform woven membrane, with distance \(h\) between adjacent strands, differ from those of the corresponding continuous membrane by less than \(c(M)h^2\), where \(c(M)\) depends on \(M\) but not on \(h\). Other results include a variational principle, theorems on existence and properties of Green's function, and a theorem on differential inequalities on networks, which has as a corollary a network analogue of the Harnack inequality for elliptic problems on manifolds.
Mathematics Subject Classification ID: 34B45
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34A40
zbMATH DE Number: 5000643
zbMATH Keywords: Sturm--Liouville problems on networks
zbMATH Keywords: eigenvalues
zbMATH Keywords: vibrations of woven membranes
zbMATH Keywords: Harnack inequality
reviewed by: Alan L. Andrew
MaRDI profile type: Publication
full work available at URL: s10958-005-0394-6
OpenAlex ID: W2037688819
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DOI: 10.1007/S10958-005-0394-6
description: scientific article; zbMATH DE number 5000643