Open Access
VOL. 5 | 1984 Spectral properties of certain stiff problems in elasticity and acoustics, part II
G. Geymonat, E. Sanchez-Palencia

Editor(s) Brian Jefferies, Alan McIntosh

Proc. Centre Math. Appl., 1984: 15-38 (1984)


We consider the vibration problem for an elastic bounded body with small compressibility, which is associated with a small parameter $\epsilon$, As $\epsilon \downarrow 0$ this is a stiff perturbation problem with non-analytic character, (in particular, the domain of the operator for $\epsilon = 0$ is not dense in a standard space, rdhereas for $\epsilon \neq 0$ it is). Nevertheless, analytic perturbation theory applies and we prove that the solution corresponding to each point of the resolvent set of the $\epsilon = 0$ problem may be expanded as a series convergent: for small $| \epsilon |$ moreover, eigenvalues and eigenvectors have holomorphic expansions for small $| \epsilon |$. Explicit computation of the first terms of the perturbation is given. The asymptotic behaviour of eigenvalues for large values of the spectral parameter is also given, and we show that it is not holomorphic in $ \epsilon $. The preceding techniques are applied to the problem of vibrations of a slightly viscous compressible fluid in a bounded vessel; an implicit function argument allows us to prove that infinitely many real eigenvalues converge as $ \epsilon + 0$ in an analytic way to the origin which is an eigenvalue of infinite multiplicity of the problem for $ \epsilon = 0$.


Published: 1 January 1984
First available in Project Euclid: 18 November 2014

zbMATH: 0597.73004
MathSciNet: MR757565

Rights: Copyright © 1984, Centre for Mathematical Analysis, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.


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