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Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 3, Pages 41–86 (Mi izv8238)  

This article is cited in 2 scientific papers (total in 2 papers)

Eigenvibrations of thick cascade junctions with ‘very heavy’ concentrated masses

T. A. Mel'nika, G. A. Chechkinb

a National Taras Shevchenko University of Kyiv, The Faculty of Mechanics and Mathematics
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We study small-parameter asymptotics of eigenelements of a boundary-value problem for the Laplace operator in a thick cascade junction with concentrated masses. There are five qualitatively different cases in the asymptotic behaviour of eigenvalues and eigenfunctions as the small parameter tends to zero (‘light’, ‘intermediate’, ‘slightly heavy’, ‘intermediate heavy’ and ‘very heavy’ concentrated masses). We study the influence of concentrated masses on the asymptotics of eigenvibrations in the last two cases. We construct the leading terms of asymptotic expansions for eigenfunctions and eigenvalues and rigorously justify them by appropriate asymptotic estimates. We also find new types of high-frequency eigenvibrations.

Keywords: thick cascade junction, concentrated masses, homogenization, matching of asymptotic expansions, eigenfunctions, eigenvalues, problems with a small parameter.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-07920
The second author's work was partially supported by RFBR (grant no. 15-01-07920).


DOI: https://doi.org/10.4213/im8238

Full text: PDF file (913 kB)
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English version:
Izvestiya: Mathematics, 2015, 79:3, 467–511

Bibliographic databases:

UDC: 517.956.225+517.956.8
MSC: 35B27, 35B40, 35J25, 35P20, 47A45, 74K30
Received: 07.04.2014
Revised: 14.11.2014

Citation: T. A. Mel'nik, G. A. Chechkin, “Eigenvibrations of thick cascade junctions with ‘very heavy’ concentrated masses”, Izv. RAN. Ser. Mat., 79:3 (2015), 41–86; Izv. Math., 79:3 (2015), 467–511

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. T. A. Mel'nyk, I. A. Nakvasiuk, “Homogenization of a semilinear variational inequality in a thick multi-level junction”, Journal of Inequalities and Applications, 104 (2016)  mathscinet  zmath  isi
    2. G. A. Chechkin, T. P. Chechkina, “Asymptotic behavior of the spectrum of an elliptic problem in a domain with aperiodically distributed concentrated masses”, C. R. Mecanique, 345:10 (2017), 671–677  crossref  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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