*Accepted Paper*

**Inserted:** 23 aug 2019

**Last Updated:** 1 nov 2020

**Journal:** Analysis & PDE

**Year:** 2019

**Abstract:**

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.

**Keywords:**
Least gradient problem - Beckmann problem - Transport density - regularity

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