Inserted: 9 jul 2022
Last Updated: 2 apr 2023
We consider a region $\Omega$ where a mass $f$ is transported to the boundary and the aim is to find an optimal free transport region $E$ that minimizes the total cost outside $E$ of this transport problem plus a penalization term on $E$. First, we study the regularity of the transport density $\sigma$ in this transport problem to the boundary. Then, we show existence of an optimal set $E$ for this shape optimization problem and, we prove regularity on this optimal set $E$ in the case where the penalization term on $E$ is given by the perimeter (or the fractional perimeter) of $E$.
Keywords: Optimal transport, regularity, Transport density, shape optimization