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

$L^p$ bounds for boundary-to-boundary transport densities, and $W^{1,p}$ bounds for the BV least gradient problem in 2D