Optimal Transportation with Boundary Costs and Summability Estimates on the Transport Density

2016/09/27 by Samer Dweik, Dweik, Samer
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #math.AP #math.FA #math.OC

paper · pdf · doi:10.48550/arxiv.1609.08418

arxiv created 2016/09/27 · openalex publication_date 2016/09/27 · arxiv updated 2016/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we analyze a mass transportation problem in a bounded domain with the possibility to transport mass to/from the boundary, paying the transport cost, that is given by the Euclidean distance plus an extra cost depending on the exit/entrance point. This problem appears in import/export model, as well as in some shape optimization problems. We study the L^p summability of the transport density which does not follow from standard theorems, as the target measures are not absolutely continuous but they have some parts of them which are concentrated on the boundary. We also provide the relevant duality arguments to connect the corresponding Beckmann and Kantorovich problems to a formulation with Kantorovich potentials with Dirichlet boundary conditions.

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