Invertibility and non-invertibility in thin elastic structures

The nonlinear elastic energy of a thin film of thickness h is given by a functional E (h) . Friesecke, James and Muller derived the I"-limits, as h -> 0, of the functionals h (-alpha) E (h) for alpha a parts per thousand 3. In this article we study the invertibility properties of almost minimizers of these functionals, and more generally of sequences with equiintegrable energy density. We show that they are invertible almost everywhere away from a thin boundary layer near the film surface. Moreover, we obtain an upper bound for the width of this layer and a uniform upper bound on the diameter of preimages. We construct examples showing that these bounds are sharp. In particular, for all alpha a parts per thousand 3 there exist Lipschitz continuous low energy deformations which are not locally invertible.