Extremal eigenvalue problems for composite membranes. I
type: Item
instance of: scholarly article
title: Extremal eigenvalue problems for composite membranes. I
zbMATH Open document ID: 0709.73044
DOI: 10.1007/BF01447325
author: Steven J. Cox
author: Joyce McLaughlin
published in: Applied Mathematics and Optimization
publication date: 1990-01-01T00:00:00Z
review text: The class of structures that is considered here are membranes on \(\Omega \in {\mathbb{R}}^ N\) with densities in \(ad_{\gamma}=\{\alpha \chi +\beta (1-\chi):\chi\) is the characteristic function of a subset of \(\Omega\) with measure \(\gamma\), \(\alpha\) and \(\beta\) are given\(\}\). The investigated composite membranes are such that their material density is \(\alpha\) in volume fraction \(\gamma /_{| \Omega |}\) and \(\beta\) in volume fraction \(1-\gamma /_{| \Omega |}\). An optimum problem is formulated for the clamped membranes, the cost of membrane with density \(\rho \in ad_{\gamma}\) is measured in terms of \(\lambda_ k(\rho)\), the membrane kth eigenvalue. This paper establishes a topology for \(ad_{\gamma}\) and studies the continuity of \(\rho \to \lambda_ k(\rho)\) on the associated closure of \(ad_{\gamma}\). Existence of extremizers for \(\lambda_ k\) is derived. Rayleigh's principle, Minmax principle, Auchmuty's principle are used in combination with convex analysis to prove some properties of the largest or smallest kth natural frequency. The obtained results are the generalizations of Krein's one- dimensional work to the N dimensional case.
Mathematics Subject Classification ID: 74P99
Mathematics Subject Classification ID: 74E30
Mathematics Subject Classification ID: 74S30
Mathematics Subject Classification ID: 74P10
Mathematics Subject Classification ID: 49R50
Mathematics Subject Classification ID: 49J27
zbMATH DE Number: 4167168
zbMATH Keywords: clamped membranes
zbMATH Keywords: Rayleigh's principle
zbMATH Keywords: Minmax principle
zbMATH Keywords: Auchmuty's principle
reviewed by: István Ecsedi
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01447325
OpenAlex ID: W4238877195
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description: scientific article; zbMATH DE number 4167168