  We show the existence of global minimizers for a geometrically nonlinear
isotropic elastic Cosserat 6-parameter shell model. The proof of the main
theorem is based on the direct methods of the calculus of variations using
essentially the convexity of the energy in the nonlinear strain and curvature
measures. We first show the existence of the solution for the theory including
$O(h^5)$ terms. The energy allows us to show the coercivity for terms up to
order $O(h^5)$ and the convexity of the energy. Secondly, we consider only that
part of the energy including $O(h^3)$ terms. In this case the obtained
minimization problem is not the same as those previously considered in the
literature, since the influence of the curved initial shell configuration
appears explicitly in the expression of the coefficients of the energies for
the reduced two-dimensional variational problem and additional mixed
bending-curvature and curvature terms are present. While in the theory
including $O(h^5)$ the conditions on the thickness $h$ are those considered in
the modelling process and they are independent of the constitutive parameter,
in the $O(h^3)$-case the coercivity is proven on some more restrictive
conditions under the thickness $h$.
