ESTIMATES FOR THE SECOND ORDER DERIVATIVES OF EIGENVECTORS IN THIN ANISOTROPIC PLATES WITH VARIABLE THICKNESS

摘要

For the second order derivatives of eigenvectors in a thin anisotropic heterogeneous plate Ω_h, we derive estimates of their weighted L_2-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 u~1 of a similar boundary-value problem for a scalar second order differential operator with variable coefficients, the norm ‖▽_x~2u~0; L_2(Ω_h)‖ is of order h~(-1) and grows as h tends to zero.