Accepted Paper: Journal de mathématiques pures et appliquées
Inserted: 11 apr 2017
Last Updated: 5 feb 2018
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 $p$.
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