  In this work we derive asymptotically sharp weighted Korn and Korn-like
interpolation (or first and a half) inequalities in thin domains with singular
weights. The constants $K$ (Korn's constant) in the inequalities depend on the
domain thickness $h$ according to a power rule $K=Ch^\alpha,$ where $C>0$ and
$\alpha\in R$ are constants independent of $h$ and the displacement field. The
sharpness of the estimates is understood in the sense that the asymptotics
$h^\alpha$ is optimal as $h\to 0.$ The choice of the weights is motivated by
several factors, in particular a spacial case occurs when making Cartesian to
polar change of variables in two dimensions.
