  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 $h\ll1$.
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 $|\ln
h|^{-1/2}$.
