Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction

Abstract

A spectral boundary-value problem is considered in a plane thick two-level junction $\Omega_{\varepsilon}$, which is the union of a domain $\Omega_{0}$ and a large number $2N$ of thin rods with thickness of order $\varepsilon = \mathcal{O} (N^{-1})$. The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are $\varepsilon$-periodically alternated. The Fourier conditions are given on the lateral boundaries of the thin rods. The asymptotic behavior of the eigenvalues and eigenfunctions is investigated as $\varepsilon \rightarrow 0$, i.e., when the number of the thin rods infinitely increases and their thickness tends to zero. The Hausdorff convergence of the spectrum is proved as $\varepsilon \rightarrow 0$, the leading terms of asymptotics are constructed and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions.