BV estimates on the transport density with Dirichlet region on the boundary

In this paper, we prove BV regularity on the transport density in the mass transport problem to the boundary in two dimensions under certain conditions on the domain, the boundary cost and the mass distribution. Moreover, we show by a counter-example that the smoothness of the mass distribution, the boundary and the boundary cost does not imply that the transport density is \(W^{1,p}\) $W^{1,p}$ , for some \(p>1\) $p>1$ .