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 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 much easier to verify, 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.

在最近的论文[2]中，证明了Sobolev范数中平面微分同态的闭包由非交叉（NC）函数组成，即可以由连续的一对一函数统一逼近的函数。网格上的一项功能。此属性的一个深层简化是考虑曲线而不是网格，因此要考虑在线上不相交的函数（NCL）。由于NCL属性更容易验证，如果它们实际上重合，则将是非常肯定的，而仅NC隐含NCL才是显而易见的。我们表明，一般而言，NCL并不意味着NC，但是在附加假设下，该含义变为正确。$\det （dü）>0$ ae，这是非线性弹性中非常普遍的假设。