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

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