## $W^{1,p}$ regularity on the solution of the BV least gradient problem with Dirichlet condition on a part of the boundary

created by dweik on 10 Dec 2020
modified on 05 Jun 2022

[BibTeX]

Accepted Paper

Inserted: 10 dec 2020
Last Updated: 5 jun 2022

Journal: Nonlinear Analysis
Year: 2020

Abstract:

In this paper, we consider the BV least gradient problem with Dirichlet condition imposed on a part $\Gamma$ of the boundary $\partial\Omega$. In 2D, we show that this problem is equivalent to an optimal transport problem with Dirichlet region $\partial\Omega \backslash\Gamma$. Thanks to this equivalence, we show existence and uniqueness of a solution $u$ to this least gradient problem. Then, we prove $W^{1,p}$ regularity on this solution $u$ by studying the $L^p$ summability of the transport density in the corresponding equivalent optimal transport formulation.