W1, p regularity on the solution of the BV least gradient problem with Dirichlet condition on a part of the boundary. (October 2022)

• Record Type: Journal Article
• Authors: Dweik, Samer
• Is Part Of: Nonlinear analysis. Volume 223(2022)
• Publication Date: 2022-10
• Subjects: 35J20 -- 35J25 -- 35J75 -- 35J92; Least gradient problem -- Optimal transport -- Dirichlet region; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.113012
• Languages: English
• ISSNs: 0362-546X

Abstract

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.