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

Article type: Research Article

Authors: Khrabustovskyi, Andrii^{; *} | Plum, Michael

Affiliations: Institute for Analysis, Department of Mathematics, Karlsruhe Institute of Technology, Englerstraße 2, 76131 Karlsruhe, Germany. E-mails: andrii.khrabustovskyi@kit.edu, michael.plum@kit.edu

Correspondence: [*] Corresponding author. Tel.: +49 721 608 42064; Fax: +49 721 608 46530; E-mail: andrii.khrabustovskyi@kit.edu.

Abstract: In this paper we study the asymptotic behaviour as ε→0 of the spectrum of the elliptic operator Aε=−1bεdiv(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, 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.

Keywords: high-contrast coefficients, spectrum asymptotics, homogenization, periodic waveguides, spectral gaps

DOI: 10.3233/ASY-161363

Journal: Asymptotic Analysis, vol. 98, no. 1-2, pp. 91-130, 2016