By a result of John Ball (1981), a locally orientation preserving Sobolev map is almost everywhere globally invertible whenever its boundary values admit a homeomorphic extension. As shown here for any dimension, the conclusions of Ball's theorem and related results can be reached while completely avoiding the problem of homeomorphic extension. For suitable domains, it is enough to know that the trace is invertible on the boundary or can be uniformly approximated by such maps. An application in Nonlinear Elasticity is the existence of homeomorphic minimizers with finite distortion whose boundary values are not fixed. As a tool in the proofs, strictly orientation-preserving maps and their global invertibility properties are studied from a purely topological point of view.

中文翻译：

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通过边界上或边界附近的可逆性来实现方向保持 Sobolev 地图的全局可逆性

根据 John Ball (1981) 的结果，只要边界值允许同胚扩展，保持局部方向的 Sobolev 映射几乎处处全局可逆。如此处所示，对于任何维度，都可以在完全避免同胚扩展问题的情况下得出鲍尔定理的结论和相关结果。对于合适的域，只要知道轨迹在边界上是可逆的或者可以被这样的映射一致地近似就足够了。非线性弹性中的一个应用是存在边界值不固定的具有有限失真的同胚极小值。作为证明中的工具，我们从纯拓扑的角度研究了严格保持方向的映射及其全局可逆性。