Calculus of Variations and Geometric Measure Theory

L. Brasco - G. Carlier

Congested traffic equilibria and degenerate anisotropic PDEs

created by brasco on 09 Oct 2012
modified on 27 Mar 2013


Accepted Paper

Inserted: 9 oct 2012
Last Updated: 27 mar 2013

Journal: Dyn. Games. Appl.
Pages: 17
Year: 2012


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.

Keywords: Traffic congestion, regularity, anisotropic and degenerate PDEs