Congested optimal transport:: modeling, equilibria and elliptic regularity

We review two models for optimal transport with congestion effects in a continuous setting: an Eulerian one proposed by Beckmann and a Lagrangian one introduced by Wardrop. We also introduce the concept of Wardrop equilibrium for the traffic assignment problem. We show how to obtain existence of such equilibria, by means of a convex minimization problem. Moreover, by exploiting the equivalence of the two models and the regularity theory for degenerate elliptic equations, we show how these equilibria can be characterized by means of flows of suitable vector fields.