*Preprint*

**Inserted:** 3 dec 2021

**Last Updated:** 19 jan 2024

**Year:** 2023

**Abstract:**

In this paper, we consider a mass transportation problem with transport cost given by a 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 $\sigma$ in the Monge-Kantorovich problem with Riemannian cost between two diffuse measures $f^+$ and $f^-$. Using some technical geometrical estimates on the transport rays, we will 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 $p \in [1,∞]$. Moreover, we will prove that the transport density between a diffuse measure $f^+$ and its Riemannian projection onto the boundary (so, the target measure is singular) is in $L^p(\Omega)$ provided that $f^+ \in L^p(\Omega)$ and $\Omega$ satisfies a uniform exterior ball condition. Finally, we will extend the $L^p$ estimates on the transport density $\sigma$ to the case of a transport problem with import-export taxes.

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