Estimates for second order derivatives of eigenvectors in thin anisotropic plates with variable thickness

For second order derivatives of eigenvectors in a thin anisotropic heterogeneous plate $\Omega_h$, we derive estimates of the weighted $L_2$-norms with the majorants whose dependence on both, the plate thickness $h$ and the eigenvalue number, are expressed explicitly. These estimates keep the asymptotic sharpness along the whole spectrum while, inside its low-frequency range, the majorants remain bounded as $h\to+0$. The latter is rather unexpected fact because, for the first eigenfunction $u^1$ of the alike boundary value problem for a scalar second order differential operator with variable coefficients, the norm $\Vert\nabla_x^2u^0;L_2(\Omega_h)\Vert$ is of order $h^{-1}$ and grows as $h$ vanishes.