In this talk, we study the equivalence between the weighted least gradient problem and the weighted Beckmann minimal flow problem or equivalently, the optimal transport problem with Riemannian cost. Thanks to this equivalence, we prove existence and uniqueness of a solution to the weighted least gradient problem. Then, we show L^p summability on the transport density between two singular measures in the corresponding equivalent optimal transport formulation, which implies Sobolev regularity on the solution of the weighted least gradient problem.