Hokkaido Mathematical Journal

Integral identities for Bi-Laplacian problems and their application to vibrating plates

Guang-Tsai LEI and Guang-Wen (George) PAN

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Abstract

In this paper we derive boundary integral identities for the bi-Laplacian eigenvalue problems under Dirichlet, Navier and simply-supported boundary conditions. By using these integral identities, we prove that the first eigenvalue of the eigenvalue problem under the simply-supported boundary conditions strictly increases with Poisson's ratio. In addition, we establish the boundary integral expressions for the strain energy calculation of the vibrating plates under the three types of boundary conditions.

Article information

Source
Hokkaido Math. J., Volume 42, Number 3 (2013), 425-443.

Dates
First available in Project Euclid: 12 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1384273391

Digital Object Identifier
doi:10.14492/hokmj/1384273391

Mathematical Reviews number (MathSciNet)
MR3137394

Zentralblatt MATH identifier
1287.35051

Subjects
Primary: 35J40: Boundary value problems for higher-order elliptic equations

Keywords
Bi-Laplacian eigenvalue problems Dirichlet boundary conditions Simply-supported boundary conditions Rellich's identity Pohozaev's identity Vibrating plates Poisson's ratio Rayleigh's conjecture

Citation

LEI, Guang-Tsai; PAN, Guang-Wen (George). Integral identities for Bi-Laplacian problems and their application to vibrating plates. Hokkaido Math. J. 42 (2013), no. 3, 425--443. doi:10.14492/hokmj/1384273391. https://projecteuclid.org/euclid.hokmj/1384273391


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