Calculus of Variations and Geometric Measure Theory

S. Dweik - W. Górny

Least gradient problem on annuli

created by dweik on 23 Aug 2019
modified on 01 Nov 2020


Accepted Paper

Inserted: 23 aug 2019
Last Updated: 1 nov 2020

Journal: Analysis & PDE
Year: 2019


We consider the two dimensional BV least gradient problem on an annulus with given boundary data $g \in BV(\partial\Omega)$. 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 - Beckmann problem - Transport density - regularity