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In this paper we consider the mass transportation problem in a bounded 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 onthe boundary f -- = P \# f + (P being the projection on the boundary), hencesingular. Via a symmetrization trick, as soon as $\Omega$ is convex orsatisfies a uniform exterior ball condition, we prove L p estimates (if f +$\in$ L p, then $\sigma$ $\in$ L p). Finally, by a counterexample we prove thatif f + $\in$ L $\infty$ $(\Omega)$ and f -- has bounded density w.r.t. thesurface measure on $\partial\Omega$, the transport density $\sigma$ between f +and f -- is not necessarily in L $\infty$ $(\Omega)$, which means that the factthat f -- = P \# f + is crucial.

Summability estimates on transport densities with dirichlet regions on the boundary via symmetrization techniques