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

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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.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 195–216, February, 2006.

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Mel’nyk, T.A. Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction. Ukr Math J 58, 220–243 (2006). https://doi.org/10.1007/s11253-006-0063-z

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  • DOI: https://doi.org/10.1007/s11253-006-0063-z

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