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

S. A. Nazarov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 33, Tome 308 (2004), pp. 161-181

Résumé

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.