Regularity of intrinsically convex $W^{2,2}$ surfaces and a derivation of a homogenized bending theory of convex shells

Abstract

We prove smoothness of $W^{2,2}$ isometric immersions of surfaces endowed with a smooth Riemannian metric of positive Gauss curvature. We then derive the $\Gamma$-limit of three dimensional nonlinear shells with inhomogeneous energy density, in the bending energy regime.