The authors prove a regularity property for certain isometric immersions of surfaces endowed with a smooth Riemannian metric of positive Gauss curvature, which arise in thin film elasticity. As an application, a homogenized nonlinear bending theory is derived for shells in three-dimensional elasticity. In the first part of the work, the authors show in Theorem 2.1 that under square integrability of the second fundamental form of an immersion, the smoothness of the metric yields the smoothness of the immersion, provided this latter belongs to the Sobolev space \(W^{2,2}\). This theorem is then used to prove the ellipticity and the existence of solution for a nonlinear, first-order partial differential equation arising in problems of thin film elasticity. The authors then derive a limit of three-dimensional nonlinear shells with inhomogeneous energy density in the bending energy regime, when the thickness of the shell and the oscillation period of the stored energy function both tend to zero.
In a second part, the authors derive a homogenized nonlinear bending theory in von Karman’s energy regime. Simultaneous homogenization and dimension reduction are used. Attention is restricted to convex shells. The starting point is the energy functional of three-dimensional nonlinear elasticity. Theorem 3.2 is the main result for this part of the work. The authors derive upper and lower bounds for the energy functional. As stated in the introduction, “the derivation of the lower bound is quite natural. However, as usual, we can prove sharpness of the lower bound only for regular limiting deformations. We are not able to close this regularity gap. However, our regularity result Theorem 2.1 allows us to narrow the gap.”