Calculus of Variations and Geometric Measure Theory

D. Campbell - A. Pratelli - E. Radici

Comparison between non-crossing and non-crossing on lines properties

created by radici on 10 Apr 2020
modified by pratelli on 20 Jan 2022


Published Paper

Inserted: 10 apr 2020
Last Updated: 20 jan 2022

Journal: Journal of Mathematical Analysis and Applications
Year: 2021


In the recent paper, 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