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 Mat. Sb., 2019, Volume 210, Number 4, Pages 3–26 (Mi msb9008)

Eigenvalue asymptotics of long Kirchhoff plates with clamped edges

F. L. Bakharev, S. A. Nazarov

Faculty of Mathematics and Mechanics, St Petersburg State University, St Petersburg, Russia

Abstract: Asymptotic expansions are constructed for the eigenvalues and eigenfunctions of the Dirichlet problem for the biharmonic operator in thin domains (Kirchhoff plates with clamped edges). For a rectangular plate the leading terms are asymptotically determined from the Dirichlet problem for a second-order ordinary differential equation, while for a $\mathsf T$-junction of plates they are determined from another limiting problem in an infinite waveguide formed by three half-strips in the shape of a letter $\mathsf T$ and describing a boundary-layer phenomenon. Open questions are stated for which the method developed gives no answer.
Bibliography: 33 titles.

Keywords: Kirchhoff plate, eigenvalues and eigenfunctions, asymptotic behaviour, dimension reduction, boundary layer.

 Funding Agency Grant Number Russian Science Foundation 17-11-01003 This research was supported by the Russian Science Foundation (project no. 17-11-01003).

DOI: https://doi.org/10.4213/sm9008

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English version:
Sbornik: Mathematics, 2019, 210:4, 473–494

Bibliographic databases:

UDC: 517.956.8+517.956.227+517.958:539.3(5)
MSC: Primary 35P20, 74K30; Secondary 35Q74

Citation: F. L. Bakharev, S. A. Nazarov, “Eigenvalue asymptotics of long Kirchhoff plates with clamped edges”, Mat. Sb., 210:4 (2019), 3–26; Sb. Math., 210:4 (2019), 473–494

Citation in format AMSBIB
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