W1,p regularity on the solution of the BV least gradient problem with Dirichlet condition on a part of the boundary

In this paper, we consider the BV least gradient problem with Dirichlet condition imposed on a part Gamma of the boundary partial derivative Omega. In 2D, we show that this problem is equivalent to an optimal transport problem with Dirichlet region partial derivative Omega\Gamma. Thanks to this equivalence, we show existence and uniqueness of a solution u to this least gradient problem. Then, we prove W-1,W-p regularity on this solution u by studying the L-p summability of the transport density in the corresponding equivalent optimal transport formulation. (c) 2022 Published by Elsevier Ltd.

Automated summary

In this paper, we study the BV least gradient problem with Dirichlet condition imposed on a part of the boundary. We establish the equivalence between this problem and an optimal transport problem. By analyzing the summability of the transport density, we prove the existence, uniqueness, and regularity of the solution to the least gradient problem.