Abstract

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 LpL^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 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.

Author information

Contact details are reproduced from the original publication and may be historical.

Samer Dweik

Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay, 91405 Orsay Cedex, France
and: Université Paris-Sud
and: CNRS

Samer.Dweik@math.u-psud.fr

Keywords

  • Optimal transport
  • Monge-Kantorovich equation
  • transport density

Mathematics Subject Classification

35B65, 46N10, 49N60

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Suggested citation

S. Dweik. “Optimal Transportation with Boundary Costs and Summability Estimates on the Transport Density.” Journal of Convex Analysis 25 (2018), No. 1, 135–160. https://doi.org/10.68381/jca25009

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Published by Heldermann Verlag, 2018. Rights now held by Banach Press LLC.