  We consider the two dimensional BV least gradient problem on an annulus with
given boundary data $g \in BV(\partial\Omega)$. Firstly, we prove that this
problem is equivalent to the optimal transport problem with source and target
measures located on the boundary of the domain. Then, under some admissibility
conditions on the trace, we show that there exists a unique solution for the BV
least gradient problem. Moreover, we prove some $L^p$ estimates on the
corresponding minimal flow of the Beckmann problem, which implies directly
$W^{1,p}$ regularity for the solution of the BV least gradient problem.
