  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 identities, we first obtain the uniqueness criteria
for the solutions of the bi-Laplacian eigenvalue problems, and then prove that
each eigenvalue of the problem with simply-supported boundary condition
increases strictly with Poisson's ratio, thereby showing that each natural
frequency of a simply-supported vibrating plate increases strictly with
Poisson's ratio. In addition, we obtain boundary integral representations for
the strain energies of the vibrating plates under the three boundary
conditions.
