# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 42 (2013) pp.425-443

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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.

MSC(Primary) 35J40 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