We define a class of Sobolev W (Ω,R) functions, with p > n− 1, such that its trace on ∂Ω is also Sobolev, and do not present cavitation in the interior or on the boundary. We show that if a function in this class has positive Jacobian and coincides on the boundary with an injective map, then the function is itself injective. We then prove the existence of minimizers within this class for the type of functionals that appear in nonlinear elasticity.

中文翻译：

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Sobolev 地图的全局可逆性

我们定义了一类 Sobolev W (Ω,R) 函数，其中 p > n− 1，这样它在∂Ω 上的轨迹也是 Sobolev，并且不会在内部或边界上出现空化。我们表明，如果此类中的函数具有正雅可比行列式并且在边界上与单射映射重合，则该函数本身就是单射的。然后，我们证明了此类中对于出现在非线性弹性中的泛函类型的最小化器的存在。