Optimal transport approach to Sobolev regularity of solutions to the weighted least gradient problem

2021/12/27 by Samer Dweik, Dweik, Samer, Wojciech Górny +1
Mathematics · Medicine · #35J25 #35J75 #49Q22 #Analysis of PDEs (math.AP) #Applied mathematics #Balanced flow #Equivalence (formal languages) #FOS: Mathematics #Mathematical analysis #Mathematical optimization #Mathematics #Nonlinear Partial Differential Equations #Orthopaedic implants and arthroplasty #Pure mathematics #Sobolev space #Uniqueness #math.AP #msc:35J25 #msc:35J75 #msc:49Q22

paper · pdf · doi:10.48550/arxiv.2112.13920

25 pages

arxiv created 2021/12/27 · openalex publication_date 2021/12/27 · arxiv updated 2021/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

We study the equivalence between the weighted least gradient problem and the weighted Beckmann minimal flow problem or equivalently, the optimal transport problem with Riemannian cost. Thanks to this equivalence, we prove existence and uniqueness of a solution to the weighted least gradient problem. Then, we show L^p regularity on the transport density between two singular measures in the corresponding equivalent Riemannian optimal transport formulation. This will imply W^1,p regularity of the solution of the weighted least gradient problem.

Related

• Applications of optimal transport methods in the least gradient problem
• L^p bounds for boundary-to-boundary transport densities, and W^1,p bounds for the BV least gradient problem in 2D
• Least gradient problem on annuli
• L^α-Regularization of the Beckmann Problem
• Deriving the Gradients of Some Popular Optimal Transport Algorithms