Calculus of Variations and Geometric Measure Theory

S. Dweik

Regularity of the transport density in the optimal transport problem with Riemannian cost

created by dweik on 03 Dec 2021
modified on 31 Mar 2023



Inserted: 3 dec 2021
Last Updated: 31 mar 2023

Year: 2021


In this paper, we consider a mass transportation problem with transport cost given by a smooth positive Riemannian metric in a bounded domain $\Omega$ where a mass $f^+$ is sent to a location $f^−$ in $\Omega$ (with the possibility of importing or exporting masses from or to the boundary $\partial\Omega$). First, we study the $L^p$ summability of the transport density $\sigma$ between two regular measures $f^+$ and $f^−$. By a geometrical proof, we show that $\sigma$ belongs to $L^p(\Omega)$ as soon as the source measure $f^+$ and the target one $f^−$ are both in $L^p(\Omega)$, for all $p$. Moreover, we prove that the transport density in the transport problem to the boundary (i.e. between a mass $f^+$ and its Riemannian projection onto the boundary, so the target measure is singular) is in $L^p(\Omega)$ provided $f^+ \in L^p(\Omega)$ and $\Omega$ satisfies a uniform exterior ball condition. Finally, we collect these results to obtain $L^p$ estimates on the transport density in the Riemannian import-export transport problem.