  An asymptotic analysis is performed for thin anisotropic elastic plate
clamped along its lateral side and also supported at a small area $\theta_{h}$
of one base with diameter of the same order as the plate thickness $h\ll1.$ A
three-dimensional boundary layer in the vicinity of the support $\theta_{h}$ is
involved into the asymptotic form which is justified by means of the previously
derived weighted inequality of Korn's type provides an error estimate with the
bound $ ch^{1/2} \left | \ln h\right| .$ Ignoring this boundary layer effect
reduces the precision order down to $\left| \ln h\right| ^{-1/2}.$ A
two-dimensional variational-asymptotic model of the plate is proposed within
the theory of self-adjoint extensions of differential operators. The only
characteristics of the boundary layer, namely the elastic logarithmic potential
matrix of size $4\times4,$ is involved into the model which however keeps the
precision order $h^{1/2}\left| \ln h\right| $ in certain norms. Several
formulations and applications of the model are discussed.
