On the Korn interpolation and second inequalities in thin domains

We consider shells of non-constant thickness in three dimensional Euclidean space around surfaces which have bounded principal curvatures. We derive Korn's interpolation (or the so called first and a half (The inequality first introduced in [Gra.Har.1])) and second inequalities on that kind of domains for $\Bu\in H^1$ vector fields, imposing no boundary or normalization conditions on $\Bu.$ The constants in the estimates are asymptotically optimal in terms of the domain thickness $h,$ with the leading order constant having the scaling $h$ as $h\to 0.$ This is the first work that determines the asymptotics of the optimal constant in the classical Korn second inequality for shells in terms of the domain thickness in almost full generality, the inequality being fulfilled for practically all thin domains $\Omega\in\mathbb R^3$ and all vector fields $\Bu\in H^1(\Omega).$ Moreover, the Korn interpolation inequality is stronger than Korn's second inequality, and it reduces the problem of estimating the gradient $\nabla\Bu$ in terms of the symmetrized gradient $e(\Bu)$, in particular any linear geometric rigidity estimates for thin domains, to the easier problem of proving the corresponding Poincar\'e-like estimates on the field $\Bu$ itself.