Optimal transport approach to Sobolev regularity of solutions to the weighted least gradient problem

TL;DR

This paper establishes a connection between the weighted least gradient problem and optimal transport, proving existence, uniqueness, and Sobolev regularity of solutions through Riemannian cost formulations.

Contribution

It introduces an equivalence between the weighted least gradient problem and Riemannian optimal transport, enabling new regularity results for solutions.

Findings

Proves existence and uniqueness of solutions.

Establishes $W^{1,p}$ regularity for solutions.

Demonstrates $L^p$ regularity of transport density between singular measures.

Abstract

We study the equivalence between the weighted least gradient problem and the weighted Beckmann minimal flow problem or equivalently, the optimal transport problem with Riemannian cost. Thanks to this equivalence, we prove existence and uniqueness of a solution to the weighted least gradient problem. Then, we show $L^p$ regularity on the transport density between two singular measures in the corresponding equivalent Riemannian optimal transport formulation. This will imply $W^{1,p}$ regularity of the solution of the weighted least gradient problem.