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

In this paper we consider the mass transportation problem in a bounded domain $Ω$ where a positive mass f + in the interior is sent to the boundary $\partialΩ$, appearing for instance in some shape optimization problems, and we prove summability estimates on the associated transport density $σ$, 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 $Ω$ is convex or satisfies a uniform exterior ball condition, we prove L p estimates (if f + $\in$ L p, then $σ$ $\in$ L p). Finally, by a counterexample we prove that if f + $\in$ L $\infty$ $(Ω)$ and f -- has bounded density w.r.t. the surface measure on $\partialΩ$, the transport density $σ$ between f + and f -- is not necessarily in L $\infty$ $(Ω)$, which means that the fact that f -- = P \# f + is crucial.