Comparison between the noncrossing and the noncrossing on lines properties
Abstract
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 noncrossing (NC), i.e., the functions which can be uniformly approximated by continuous onetoone functions on the grids. A deep simplification of this property is to consider curves instead of grids, so considering functions which are noncrossing 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.08252
 Bibcode:
 2020arXiv200408252C
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Metric Geometry