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. Heterog. 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 as in Beckmann (Econometrica 20:643–660, 1952 ). 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. Copyright Springer Science+Business Media New York 2013
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