Equilibrium of thin shells under large strains without through-the-thickness shear and self-penetration of matter

We consider elastic thin shells without through-the-thickness shear and depict them as Gauss graphs of parametric surfaces. (We use the term shells to include plates and thin films therein.) We consider an energy depending on the first derivative of the Gauss map (so, it involves curvatures) and its second-rank minors. For it we prove existence of minimizers in terms of currents carried by Gauss graphs. In the limiting process we adopt sequences of competitors that satisfy a condition that prevents self-penetration of matter.