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

  • T. A. Mel'nik Київ. нац. ун-т iм. Т. Шевченка

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.
Published
25.02.2006
How to Cite
Mel’nik, T. A. “Asymptotic Behavior of Eigenvalues and Eigenfunctions of the Fourier Problem in a Thick Multilevel Junction”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, no. 2, Feb. 2006, pp. 195–216, http://umj.imath.kiev.ua/index.php/umj/article/view/3447.
Section
Research articles