# Lack of regularity of the transport density in the Monge problem

created by dweik on 11 Apr 2017
modified on 24 Apr 2017

[BibTeX]

Preprint

Inserted: 11 apr 2017
Last Updated: 24 apr 2017

Year: 2017

Abstract:

In this paper, we provide a family of counter-examples to the regularity of the transport density in the classical Monge-Kantorovich problem. We prove that the $W^{1,p}$ regularity of the source and target measures $f^\pm$ does not imply that the transport density $\sigma$ is $W^{1,p}$, that the $BV$ regularity of $f^\pm$ does not imply that $\sigma$ is $BV$ and that $f^\pm \in C^\infty$ does not imply that $\sigma$ is $W^{1,p}$, for large $p$.