  The least gradient problem (minimizing the total variation with given
boundary data) is equivalent, in the plane, to the Beckmann minimal-flow
problem with source and target measures located on the boundary of the domain,
which is in turn related to an optimal transport problem. Motivated by this
fact, we prove L p summability results for the solution of the Beckmann problem
in this setting, which improve upon previous results where the measures were
themselves supposed to be L p. In the plane, we carry out all the analysis for
general strictly convex norms, which requires to first introduce the
corresponding optimal transport tools. We then obtain results about the W 1,p
regularity of the solution of the anisotropic least gradient problem in
uniformly convex domains.
