# Absence of Critical Mass Densities for a Vibrating Membrane

Absence of Critical Mass Densities for a Vibrating Membrane We study the dependence of the eigenvalues of a N -dimensional vibrating membrane upon variation of the mass density. We prove that the elementary symmetric functions of the eigenvalues depend real-analytically on the mass density and that such functions have no critical points with constant mass constraint. In particular, the elementary symmetric functions of the eigenvalues, hence all simple eigenvalues, have no local maxima or minima on the set of those mass densities with a prescribed total mass. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Absence of Critical Mass Densities for a Vibrating Membrane

, Volume 59 (3) – Jun 1, 2009
9 pages

/lp/springer_journal/absence-of-critical-mass-densities-for-a-vibrating-membrane-YSK5g5wS8H
Publisher
Springer-Verlag
Subject
Mathematics; Numerical and Computational Methods ; Mathematical Methods in Physics; Mathematical and Computational Physics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-008-9055-8
Publisher site
See Article on Publisher Site

### Abstract

We study the dependence of the eigenvalues of a N -dimensional vibrating membrane upon variation of the mass density. We prove that the elementary symmetric functions of the eigenvalues depend real-analytically on the mass density and that such functions have no critical points with constant mass constraint. In particular, the elementary symmetric functions of the eigenvalues, hence all simple eigenvalues, have no local maxima or minima on the set of those mass densities with a prescribed total mass.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jun 1, 2009

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