*Preprint*

**Inserted:** 3 dec 2021

**Last Updated:** 24 dec 2021

**Year:** 2021

**Abstract:**

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.

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