In this paper we study the asymptotic behaviour as ε→0 of the
spectrum of the elliptic operator Aε=−bε1div(aε∇) posed in a bounded domain
Ω⊂Rn(n≥2) subject to Dirichlet boundary
conditions on ∂Ω. When ε→0 both coefficients
aε and bε become high contrast in a small
neighborhood of a hyperplane Γ intersecting Ω. We prove that the
spectrum of Aε converges to the spectrum of an operator
acting in L2(Ω)⊕L2(Γ) and generated by the operation
−Δ in Ω∖Γ, the Dirichlet boundary conditions on
∂Ω and certain interface conditions on Γ containing the
spectral parameter in a nonlinear manner. The eigenvalues of this operator may
accumulate at a finite point. Then we study the same problem, when Ω is
an infinite straight strip ("waveguide") and Γ is parallel to its
boundary. We show that Aε has at least one gap in the
spectrum when ε is small enough and describe the asymptotic
behaviour of this gap as ε→0. The proofs are based on methods of
homogenization theory.Comment: In the second version we added the case r=0, also the presentation is
essentially improved. The manuscript is submitted to a journa