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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

Mathematics and Mechanics Faculty, St. Petersburg State University

Abstract: We construct asymptotic expansions of eigenvalues and eigenfunctions of the Dirichlet problem for the biharmonic operator in thin domains (Kirchhoff plates with clamped edges). For a rectangular plate the principal terms are determined asymptotically from the Dirichlet problem for second-order ordinary differential equations. For ${\mathsf T}$-shaped junction of plates these terms are determined from another limiting problem in an infinite waveguide obtained as a union of three half-strips forming a letter ${\mathsf T}$ and describing the boundary layer phenomenon. Open questions are formulated on which the developed method did not provide answers.

Keywords: Kirchhoff plate, eigenvalues and eigenfunctions, asymptotics, dimension reduction, boundary layer, one-dimensional model.

 Funding Agency Grant Number Russian Science Foundation 17-11-01003

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

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English version:
DOI: https://doi.org/10.1070/SM9008

Document Type: Article
UDC: 517.956.8+517.956.227+517.958:539.3(5)
MSC: 35J40, 35P20

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

Citation in format AMSBIB
\Bibitem{BakNaz19} \by F.~L.~Bakharev, S.~A.~Nazarov \paper Eigenvalue asymptotics of long Kirchhoff plates with clamped edges \jour Mat. Sb. \yr 2019 \vol 210 \issue 4 \pages 3--26 \mathnet{http://mi.mathnet.ru/msb9008} \crossref{https://doi.org/10.4213/sm9008} \elib{http://elibrary.ru/item.asp?id=37180599}