Calculus of Variations and Geometric Measure Theory

S. Dweik - F. Santambrogio

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

created by santambro on 05 Jun 2016
modified by dweik on 09 Oct 2018


Accepted Paper: COCV

Inserted: 5 jun 2016
Last Updated: 9 oct 2018

Year: 2017


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