Spectral Analysis in a Thin Domainwith Periodically Oscillating Characteristics

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.