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

Abstract

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