#### Vol. 15, No. 3, 2022

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Least gradient problem on annuli

### Samer Dweik and Wojciech Górny

Vol. 15 (2022), No. 3, 699–725
##### Abstract

We consider the two-dimensional BV least gradient problem on an annulus with given boundary data $g\in \mathrm{BV}\left(\partial \mathrm{\Omega }\right)$. 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.

##### Keywords
least gradient problem, optimal transport, nonconvex domains, Beckmann problem, transport density, regularity
##### Mathematical Subject Classification 2010
Primary: 35J20, 35J25, 35J75, 35J92
##### Milestones
Received: 10 December 2019
Revised: 5 May 2020
Accepted: 30 October 2020
Published: 10 June 2022
##### Authors
 Samer Dweik Department of Mathematics University of British Columbia Vancouver, BC Canada Wojciech Górny Faculty of Mathematics, Informatics and Mechanics University of Warsaw Warsaw Poland