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 BV (Ω). 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 Lp estimates on the corresponding minimal flow of the Beckmann problem, which implies directly W1,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