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

Abstract

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.