Least gradient problem on annuli

TL;DR

This paper links the two-dimensional BV least gradient problem on an annulus to optimal transport, establishing existence, uniqueness, and regularity results for solutions under certain conditions.

Contribution

It demonstrates the equivalence between the BV least gradient problem and optimal transport with boundary measures, and provides regularity estimates for solutions.

Findings

Equivalence between least gradient and optimal transport problems.

Existence and uniqueness of solutions under admissibility conditions.

L^p regularity estimates for the minimal flow.

Abstract

We consider the two dimensional BV least gradient problem on an annulus with given boundary data $g \in BV(\partial\Omega)$. Firstly, we prove that this problem is equivalent to the optimal transport problem with source and target measures located on the boundary of the domain. Then, under some admissibility conditions on the trace, we show that there exists a unique solution for the BV least gradient problem. Moreover, we prove some $L^p$ estimates on the corresponding minimal flow of the Beckmann problem, which implies directly $W^{1,p}$ regularity for the solution of the BV least gradient problem.