Spectral properties of elliptic operator with double-contrast coefficients near a hyperplane

In this paper we study the asymptotic behaviour as e ! 0 of the spectrum of the elliptic operator A e = 􀀀 1 be div(aeÑ) posed in a bounded domain W Rn (n 2) subject to Dirichlet boundary conditions on ¶W. When e !0 both coefficients ae and be become high contrast in a small neighborhood of a hyperplane G intersecting W. We prove that the spectrum of A e converges to the spectrum of an operator acting in L2(W)L2(G) and generated by the operation 􀀀D in WnG, the Dirichlet boundary conditions on ¶W and certain interface conditions on G 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 W is an infinite straight strip (“waveguide”) and G is parallel to its boundary. We show that A e has at least one gap in the spectrum when e is small enough and describe the asymptotic behaviour of this gap as e ! 0. The proofs are based on methods of homogenization theory.