The nonlinear elastic energy of a thin film of thickness h is given by a functional E h . Friesecke, James and Müller derived the Γ-limits, as h → 0, of the functionals h −α E h for α ≧ 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 α ≧ 3 there exist Lipschitz continuous low energy deformations which are not locally invertible.
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机译：厚度为h的薄膜的非线性弹性能由函数E h 给出。 Friesecke，James和Müller推导了α≥3的函数h -α E h 的Γ极限，当h→0时。在本文中，我们研究了可逆性这些功能的几乎最小化子的性质，更一般地讲，具有相等积分能量密度的序列的性质。我们表明，在靠近薄膜表面的薄边界层的几乎所有地方，它们都是可逆的。此外，我们获得了该层宽度的上限，并且获得了原像直径的统一上限。我们构建的例子表明这些界限是尖锐的。特别地，对于所有α≥3，存在Lipschitz连续的低能量变形，该变形不是局部可逆的。
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