  We consider the eigenvalue problem for the Reissner-Mindlin system arising in
the study of the free vibration modes of an elastic clamped plate. We provide
quantitative estimates for the variation of the eigenvalues upon variation of
the shape of the plate. We also prove analyticity results and establish
Hadamard-type formulas. Finally, we address the problem of minimization of the
eigenvalues in the case of isovolumetric domain perturbations. In the spirit of
the Rayleigh conjecture for the biharmonic operator, we prove that balls are
critical points with volume constraint for all simple eigenvalues and the
elementary symmetric functions of multiple eigenvalues.
