Thin Elastic Plates Supported over Small Areas. I: Korn's Inequalities and Boundary Layers

A thin anisotropic elastic plate clamped along its lateral side and also supported at a small area $\theta_{h}$ of one base is considered; the diameter of $\theta_{h}$ is of the same order as the plate relative thickness $h\ll 1$. In addition to the standard Kirchhoff model with the Sobolev point condition, a three-dimensional boundary layer is investigated in the vicinity of the support $\theta_{h}$, which with the help of the derived weighted inequality of Korn's type, will provide an error estimate with the bound $ch^{1/2}|\ln h|$. Ignoring this boundary layer effect reduces the precision order down to $|\ln h|^{-1/2}$.