Geodesic
Eigenvibrations of thick cascade junctions with `very heavy' concentrated masses
Izvestiya. Mathematics , Tome 79 (2015) no. 3, pp. 467-511

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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. 
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     author = {T. A. Mel'nik and G. A. Chechkin},
     title = {Eigenvibrations of thick cascade junctions with `very heavy' concentrated masses},
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     url = {http://geodesic.mathdoc.fr/item/IM2_2015_79_3_a2/}
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T. A. Mel'nik; G. A. Chechkin. Eigenvibrations of thick cascade junctions with `very heavy' concentrated masses. Izvestiya. Mathematics , Tome 79 (2015) no. 3, pp. 467-511. http://geodesic.mathdoc.fr/item/IM2_2015_79_3_a2/