W1, p regularity on the solution of the BV least gradient problem with Dirichlet condition on a part of the boundary. (October 2022) - Dweik, Samer
Abstract: In this paper, we consider the BV least gradient problem with Dirichlet condition imposed on a part Γ of the boundary ∂ Ω . In 2D, we show that this problem is equivalent to an optimal transport problem with Dirichlet region ∂ Ω ∖ Γ . 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.