Applications of optimal transport methods in the least gradient problem

TL;DR

This paper explores the connection between the least gradient problem and boundary-to-boundary optimal transport in 2D, extending their duality and establishing regularity and stability results using optimal transport methods.

Contribution

It extends the equivalence between the least gradient problem and optimal transport to their duals and provides new regularity and stability results.

Findings

Established duality between least gradient and optimal transport problems.

Proved regularity results for the least gradient problem.

Demonstrated stability of solutions using optimal transport techniques.

Abstract

We study the consequences of the equivalence between the least gradient problem and a boundary-to-boundary optimal transport problem in two dimensions. We extend the relationship between the two problems to their respective dual problems, as well as prove several regularity and stability results for the least gradient problem using optimal transport techniques.