*Accepted Paper: COCV*

**Inserted:** 5 jun 2016

**Last Updated:** 9 oct 2018

**Year:** 2017

**Abstract:**

In this paper we consider the mass transportation problem in a bound\-ed domain $\Omega$ where a positive mass $f^+$ in the interior is sent to the boundary $\partial\Omega$, appearing for instance in some shape optimization problems, and we prove summability estimates on the associated transport density $\sigma$, which is the transport density from a diffuse measure to a measure on the boundary $f^-=P_\#f^+$ ($P$ being the projection on the boundary), hence singular. Via a symmetrization trick, as soon as $\Omega$ is convex or satisfies a uniform exterior ball condition, we prove $L^p$ estimates (if $f^+\in L^p$, then $\sigma\in L^p$). Finally, by a counter-example we prove that if $f^+ \in L^{\infty}(\Omega)$ and $f^-$ has bounded density w.r.t. the surface measure on $\partial\Omega$, the transport density $\sigma$ between $f^+$ and $f^-$ is not necessarily in $L^{\infty}(\Omega)$, which means that the fact that $f^-=P_\#f^+$ is crucial.

**Keywords:**
optimal transport, monge-kantorovich system

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