Spectral results for membrane with perturbed stiffness and mass density
38 mins 48 secs,
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Description: |
Babych, N (Bath)
Tuesday 27 March 2007, 17:00-17:30 |
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Created: | 2008-02-19 14:50 | ||
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Collection: | Highly Oscillatory Problems: Computation, Theory and Application | ||
Publisher: | Isaac Newton Institute | ||
Copyright: | Babych, N | ||
Language: | eng (English) | ||
Credits: |
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Abstract: | We study the spectrum of a nonhomogeneous membrane that consists of two parts with strongly different stiffness and mass density. The small parameter describes the quotient of stiffness coefficients. The M-th power of the parameter is comparable to the ratio of densities. We show that the asymptotic behaviour of eigenvalues and eigenfunctions depends on rate M. We distinguish three cases M<1, M=1 and M>1.
The strong resolvent convergence of perturbed operators leads to loss of the completeness of limit eigenfunction system in both cases when M is different from 1. Therefore the limit operators describe only a part of the prelimit membrane vibrations. With an eye to close this gap we use the WKB-asymptotic expansions with a quantized small parameter to prove the existence of other kind of eigenvibrations, namely high frequency vibrations. In the critical case M=1 the limit operator is a nonself-adjoint one, nevertheless the perturbed operators are self-adjoint in a certain topology. The multiplicity of spectrum and structure of root spaces are investigated. Complete asymptotic expansions for eigenelements are constructed and justified in each case of the perturbations. |
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