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

created by dweik on 24 Dec 2021
modified on 04 Nov 2022

[BibTeX]

Accepted Paper

Inserted: 24 dec 2021
Last Updated: 4 nov 2022

Journal: SIAM Journal on Mathematical Analysis
Year: 2021

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 on the solution of the weighted least gradient problem.