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

Abstract

In this paper we study the asymptotic behaviour as e → 0 of the spectrum of the elliptic operator A e =− 1 be div(a e ∇) posed in a bounded domain � ⊂ R n (n 2) subject to Dirichlet boundary conditions on ∂� .W hene → 0 both coefficients a e and b e become high contrast in a small neighborhood of a hyperplaneintersecting � . We prove that the spectrum of A e converges to the spectrum of an operator acting in L 2 (�) ⊕ L 2 (�) and generated by the operation −� in � � , Dirichlet boundary conditions on ∂� and certain interface conditions oncontaining the spectral parameter in a nonlinear manner. The eigenvalues of this operator may accumulate at a finite point. Then we study the same problem, whenis an infinite straight strip (waveguide) andis 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.