Inserted: 31 oct 2012
Last Updated: 19 aug 2013
Journal: Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov.
This paper has been written for possible publication in the proceedings of the conference ``Monge-Kantorovich optimal transportation problem, transport metrics and their applications'', organized in St Petersburg in June 2012.
We review two models of optimal transport, where congestion effects during the transport can be possibly taken into account. The first model is Beckmann's one, where the transport activities are modeled by vector fields with given divergence. The second one is the model by Carlier et al. (SIAM J Control Optim 47: 1330--1350, 2008), which in turn is the continuous reformulation of Wardrop's model on graphs. We discuss the extensions of these models to their natural functional analytic setting and show that they are indeed equivalent, by using Smirnov decomposition theorem for normal $1-$currents.
Keywords: Monge-Kantorovich problem, flat norm, Beckmann's problem, Smirnov Theorem