Congested Traffic Equilibria and Degenerate Anisotropic PDEs

Congested traffic problems on very dense networks lead, at the limit, to minimization problems posed on measures on curves as shown in \cite{BC}. Here, we go one step further by showing that these problems can be reformulated in terms of the minimization of an integral functional over a set of vector fields with prescribed divergence. We prove a Sobolev regularity result for their minimizers despite the fact that the Euler-Lagrange equation of the dual is highly degenerate and anisotropic. This somehow extends the analysis of \cite{BCS} to the anisotropic case.