On the regularity of the transport density in the import/export transport problem

The mass transport problem dates back to a work from 1781 by Gaspard Monge (Mémoire sur la théorie des déblais et des remblais) where he formulated a natural question in economics which deals with the optimal way of moving points from one mass distribution to another in some domain so that the total work done is minimized. In such a theory, it is classical to associate with any optimal transport map/plan a positive measure (called transport density) which represents the amount of transport taking place in each region of the domain.

  In this talk, we consider this Monge-Kantorovich transport problem. First, we show that this problem has an equivalent minimal flow formulation (called the Beckmann problem). Then, we prove existence of an optimal flow for this problem and, we show regularity on this optimal flow by studying the (L^p, BV, Sobolev or Hoelder) regularity of the transport density in our transport problem between two regular distributions. 

  On the other hand, we introduce and analyse an import/export transport problem (with boundary taxes). Then, we study the L^p summability of the corresponding transport density, where the source and target distributions are now singular. Finally, we prove BV estimates on this transport density.