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

Dweik, Samer; Santambrogio, Filippo

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1167-1180.

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. This problems appears, for instance in some shape optimization issues. 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 bundary), 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 }\left(\Omega \right) 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 }\left(\Omega \right), which means that the fact that f^{-}=P_{\#}{f}^{+} is crucial.

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