Optimal transport approach to Sobolev regularity of solutions to the weighted least gradient problem

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 $L^p$ regularity on the transport density between two singular measures in the corresponding equivalent Riemannian optimal transport formulation. This will imply $W^{1,p}$ regularity of the solution of the weighted least gradient problem.