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

Abstract

In this paper we study the asymptotic behaviour as $\mathit{\epsilon }\to 0$ of the spectrum of the elliptic operator ${\mathcal{A}}^{\mathit{\epsilon }}=-\frac{1}{{\mathit{b}}^{\mathit{\epsilon }}}div\mathrm{(}{\mathit{a}}^{\mathit{\epsilon }}\nabla \mathrm{)}$ posed in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^{\mathit{n}}$ ( $\mathit{n}⩾2$ ) subject to Dirichlet boundary conditions on $\mathrm{\partial }\mathrm{\Omega }$ . When $\mathit{\epsilon }\to 0$ both coefficients ${\mathit{a}}^{\mathit{\epsilon }}$ and ${\mathit{b}}^{\mathit{\epsilon }}$ become high contrast in a small neighborhood of a hyperplane Γ intersecting Ω. We prove that the spectrum of ${\mathcal{A}}^{\mathit{\epsilon }}$ converges to the spectrum of an operator acting in ${\mathit{L}}^2\mathrm{(}\mathrm{\Omega }\mathrm{)}\oplus {\mathit{L}}^2\mathrm{(}\mathrm{\Gamma }\mathrm{)}$ and generated by the operation $-\mathrm{\Delta }$ in $\mathrm{\Omega }\setminus \mathrm{\Gamma }$ , Dirichlet boundary conditions on $\mathrm{\partial }\mathrm{\Omega }$ 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 ${\mathcal{A}}^{\mathit{\epsilon }}$ has at least one gap in the spectrum when ε is small enough and describe the asymptotic behaviour of this gap as $\mathit{\epsilon }\to 0$ . The proofs are based on methods of homogenization theory.