Calculus of Variations and Geometric Measure Theory

S. Dweik

Optimal region for the transport problem to the boundary

created by dweik on 09 Jul 2022
modified on 14 Nov 2023


Accepted Paper

Inserted: 9 jul 2022
Last Updated: 14 nov 2023

Journal: Journal of Mathematical Analysis and Applications
Year: 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