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

created by santambro on 06 Mar 2018
modified on 01 May 2018

[BibTeX]

Submitted Paper

Inserted: 6 mar 2018
Last Updated: 1 may 2018

Year: 2018
Notes:

The present one is a second version, after some useful remarks by colleagues. It includes an explicit treatment of the anisotropic case.

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.