Congested Traffic Equilibria and Degenerate Anisotropic PDEs

Abstract

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