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| We consider the eigenvalue problems for the Reissner-Mindlin system and the biharmonic equation, arising in the study of the free vibration modes of an elastic clamped plate. We discuss the interplay between the two problems with particular reference to spectral stability results and domain perturbation problems. We prove stability estimates for the variation of the eigenvalues upon variation of the domain. Our estimates are expressed in terms of the Hausdorff distance between the domains and are independent of the thickness
of the plate. We also prove analyticity results for the dependence of the eigenvalues via `transplantation' 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 both of the biharmonic equation and of the Reissner-Mindlin system. Joint work with Davide Buoso. |
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