Congested traffic equilibria and degenerate anisotropic PDEs

Lorenzo Brasco (1) , Guillaume Carlier (2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

Congested traffic problems on very dense networks lead, at the limit, to minimization problems posed on measures on curves as shown in Baillon and Carlier (Netw. Heterogenous Media 7: 219--241, 2012). 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 Brasco et al. (J. Math. Pures Appl. 93: 652--671, 2010) to the anisotropic case.

Mots clés: traffic congestion; anisotropic and degenerate PDEs; regularity

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Lorenzo Brasco, Guillaume Carlier. Congested traffic equilibria and degenerate anisotropic PDEs. 2012. ⟨hal-00734555⟩