论文标题
在线属性上的非交叉和非交叉之间的比较
Comparison between the non-crossing and the non-crossing on lines properties
论文作者
论文摘要
在最近的论文[2]中，证明了Sobolev规范中平面差异性的闭合是由非交叉的功能组成的（NC），即，在电网上连续的一对一函数可以统一地近似的功能。对此属性的深入简化是考虑曲线而不是网格，因此考虑不跨线（NCL）的函数。由于NCL属性更容易检查，因此，如果它们实际上是重合的，那将是非常积极的，而NC显然暗示了NCL。我们表明，通常NCL并不意味着NC，但是对$ \ det（du）> 0 $ a.e.的其他假设的含义变得正确，这在非线性弹性中是一个非常普遍的假设。
In the recent paper [2], it was proved that the closure of the planar diffeomorphisms in the Sobolev norm consists of the functions which are non-crossing (NC), i.e., the functions which can be uniformly approximated by continuous one-to-one functions on the grids. A deep simplification of this property is to consider curves instead of grids, so considering functions which are non-crossing on lines (NCL). Since the NCL property is way easier to check, it would be extremely positive if they actually coincide, while it is only obvious that NC implies NCL. We show that in general NCL does not imply NC, but the implication becomes true with the additional assumption that $\det(Du)>0$ a.e., which is a very common assumption in nonlinear elasticity.
