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

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 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.