Spectral analysis in a thin domain with periodically oscillating characteristics

¹ 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
² 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
³ Narvik University College, P.O. Box 385, 8505 Narvik, Norway
⁴ P.N. Lebedev Physical Institute RAS, Leninski prospect 53, Moscow 119991, Russia
andrey@sci.lebedev.ru

Received: 27 July 2010

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 = ε^τ, τ < 1), or ε is much greater than δ(δ = ε^τ, τ > 1). We consider all three cases.