  In this paper we prove asymptotically sharp weighted "first-and-a-half" $2D$
Korn and Korn-like inequalities with a singular weight occurring from Cartesian
to cylindrical change of variables. We prove some Hardy and the so-called
"harmonic function gradient separation" inequalities with the same singular
weight. Then we apply the obtained $2D$ inequalities to prove similar
inequalities for washers with thickness $h$ subject to vanishing Dirichlet
boundary conditions on the inner and outer thin faces of the washer. A washer
can be regarded in two ways: As the limit case of a conical shell when the
slope goes to zero, or as a very short hollow cylinder. While the optimal Korn
constant in the first Korn inequality for a conical shell with thickness $h$
and with a positive slope scales like $h^{1.5}$ e.g. [10], the optimal Korn
constant in the first Korn inequality for a washer scales like $h^2$ and
depends only on the outer radius of the washer as we show in the present work.
The Korn constant in the first and a half inequality scales like $h$ and
depends only on $h.$ The optimal Korn constant is realized by a Kirchoff
Ansatz. This results can be applied to calculate the critical buckling load of
a washer under in plane loads, e.g. [1].
