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Spectral analysis in a thin domain with periodically oscillating characteristics

Published online by Cambridge University Press:  22 June 2011

Rita Ferreira ,
Luísa M. Mascarenhas  and
Andrey Piatnitski
Rita Ferreira
Affiliation:
I.C.T.I. - Carnegie Mellon | Portugal, F.C.T./C.M.A. da U.N.L., Quinta da Torre, 2829–516 Caparica, Portugal. rferreir@andrew.cmu.edu; ragf@fct.unl.pt
Luísa M. Mascarenhas
Affiliation:
Departamento de Matemática da F.C.T./C.M.A. da U.N.L., Quinta da Torre, 2829–516 Caparica, Portugal; mascar@fct.unl.pt
Andrey Piatnitski
Affiliation:
Narvik University College, P.O. Box 385, 8505 Narvik, Norway P.N. Lebedev Physical Institute RAS, Leninski prospect 53, Moscow 119991, Russia; andrey@sci.lebedev.ru
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Abstract

The paper deals with a Dirichlet spectral problem for an elliptic operator with ε-periodic coefficients in a 3D bounded domain of small thickness δ. We study the asymptotic behavior of the spectrum as ε and δ tend to zero. This asymptotic behavior depends crucially on whether ε and δ are of the same order (δ ≈ ε), or ε is much less than δ(δ = ετ, τ < 1), or ε is much greater than δ(δ = ετ, τ > 1). We consider all three cases.

Keywords

Spectral analysisdimension reductionperiodic homogenizationΓ-convergenceasymptotic expansions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 2 , April 2012 , pp. 427 - 451
Copyright
© EDP Sciences, SMAI, 2011

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