A continuous model of transportation revisited

Lorenzo Brasco (1) , Mircea Petrache (2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 Departement Mathematik

We review two models of optimal transport, where congestion eff ects during the transport can be possibly taken into account. The first model is Beckmann's one, where the transport activities are modeled by vector fi elds 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, using an {\it ad hoc} generalization of Smirnov Decomposition Theorem for flat 1-currents which may fail to be normal.

Mots clés: Monge-Kantorovich problem; Beckmann's problem; Smirnov Theorem; flat norm

Domaines: Analyse fonctionnelle [math.FA]; Equations aux dérivées partielles [math.AP]

Lorenzo Brasco, Mircea Petrache. A continuous model of transportation revisited. 2012. ⟨hal-00758565v1⟩