Estimates for the Second Order Derivatives of Eigenvectors in Thin Anisotropic Plates with Variable Thickness

Abstract

For the second order derivatives of eigenvectors in a thin anisotropic heterogeneous plate Ωh, we derive estimates of their weighted L2-norms with majorants whose dependence on the plate thickness h and on the eigenvalue number is expressed explicitly. These estimates maintain the asymptotic sharpness throughout the entire spectrum, whereas inside its low-frequency band the majorants remain bounded as h → +0. The latter is a rather unexpected fact, because for the first eigenfunction u1 of a similar boundary-value problem for a scalar second order differential operator with variable coefficients, the norm ‖∇x2u0; L2(Ωh)‖ is of order h−1 and grows as h tends to zero. Bibliography: 35 titles.