Ukrainian Mathematical Journal

, Volume 58, Issue 2, pp 220–243 | Cite as

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

  • T. A. Mel’nyk


A spectral boundary-value problem is considered in a plane thick two-level junction Ωε formed as the union of a domain Ω0 and a large number 2N of thin rods with thickness of order ε = O(N−1). The thin rods are split into two levels depending on their length. In addition, the thin rods from the indicated levels are ε-periodically alternating. The Fourier conditions are given on the lateral boundaries of the thin rods. The asymptotic behavior of the eigenvalues and eigenfunctions is investigated as ε → 0, i.e., when the number of thin rods infinitely increases and their thickness approaches zero. The Hausdorff convergence of the spectrum is proved as ε → 0, the leading terms of asymptotics are constructed, and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions.


Spectral Problem Neumann Problem Essential Spectrum Joint Zone Outer Expansion 
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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • T. A. Mel’nyk
    • 1
  1. 1.Shevchenko Kyiv National UniversityKyiv

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