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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
Article

Abstract

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.

Keywords

Spectral Problem Neumann Problem Essential Spectrum Joint Zone Outer Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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