2018/05/02 by Filippo Santambrogio, Santambrogio, Filippo, Samer Dweik +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Boundary (topology) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometry #Mathematical analysis #Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Optimization and Control (math.OC) #Physics #math.AP #math.CA #math.OC
paper · pdf · doi:10.48550/arxiv.1805.00769
arxiv created 2018/05/02 · openalex publication_date 2018/05/02 · arxiv updated 2018/05/03 · openalex created_date 2018/05/07 · openalex updated_date 2026/08/05
The least gradient problem (minimizing the total variation with given boundary data) is equivalent, in the plane, to the Beckmann minimal-flow problem with source and target measures located on the boundary of the domain, which is in turn related to an optimal transport problem. Motivated by this fact, we prove L p summability results for the solution of the Beckmann problem in this setting, which improve upon previous results where the measures were themselves supposed to be L p. In the plane, we carry out all the analysis for general strictly convex norms, which requires to first introduce the corresponding optimal transport tools. We then obtain results about the W 1,p regularity of the solution of the anisotropic least gradient problem in uniformly convex domains.