  This paper is concerned with the study of linear geometric rigidity of
shallow thin domains under zero Dirichlet boundary conditions on the
displacement field on the thin edge of the domain. A shallow thin domain is a
thin domain that has in-plane dimensions of order $O(1)$ and $\epsilon,$ where
$\epsilon\in (h,1)$ is a parameter (here $h$ is the thickness of the shell).
The problem has been solved in [8,10] for the case $\epsilon=1,$ with the
outcome of the optimal constant $C\sim h^{-3/2},$ $C\sim h^{-4/3},$ and $C\sim
h^{-1}$ for parabolic, hyperbolic and elliptic thin domains respectively. We
prove in the present work that in fact there are two distinctive scaling
regimes $\epsilon\in (h,\sqrt h]$ and $\epsilon\in (\sqrt h,1),$ such that in
each of which the thin domain rigidity is given by a certain formula in $h$ and
$\epsilon.$ An interesting new phenomenon is that in the first (small
parameter) regime $\epsilon\in (h,\sqrt h]$, the rigidity does not depend on
the curvature of the thin domain mid-surface.
