  In this paper we derive the linear elastic Cosserat shell model incorporating
effects up to order $O(h^5)$ in the shell thickness $h$ as a particular case of
the recently introduced geometrically nonlinear elastic Cosserat shell model.
The existence and uniqueness of the solution is proven in suitable admissible
sets. To this end, inequalities of Korn-type for shells are established which
allow to show coercivity in the Lax-Milgram theorem. We are also showing an
existence and uniqueness result for a truncated $O(h^3)$ model. Main issue is
the suitable treatment of the curved reference configuration of the shell. Some
connections to the classical Koiter membrane-bending model are highlighted.
