  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.
