Abstract
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.
Similar content being viewed by others
Notes
In particular, the fact that the travelling time functions on arcs [x,x+εv k ] scale like ε h k (x,m/ε) and that the discrete measures \(f_{+}^{G_{\varepsilon}}\) and \(f_{-}^{G_{\varepsilon}}\) weakly converge to some f + and f −.
Interestingly, the connection with the Monge–Kantorovich theory was realized much later by Robert McCann.
References
Ambrosio L (2004) Transport equation and Cauchy problem for BV vector fields. Invent Math 158:227–260
Baillon J-B, Carlier G (2012) From discrete to continuous Wardrop equilibria. Netw Heterog Media 7:219–241
Beckmann MJ (1952) A continuous model of transportation. Econometrica 20:643–660
Beckmann M, McGuire C, Winsten C (1956) Studies in economics of transportation. Yale University Press, New Haven
Belloni M, Kawohl B (2004) The pseudo p-Laplace eigenvalue problem and viscosity solutions as p→∞. ESAIM Control Optim Calc Var 10:28–52
Brasco L, Carlier G (2012) On certain anisotropic elliptic equations arising in congested optimal transport: local gradient bounds. Preprint. http://cvgmt.sns.it/paper/1890/
Brasco L, Carlier G, Santambrogio F (2010) Congested traffic dynamics, weak flows and very degenerate elliptic equations. J Math Pures Appl 93:652–671
Canale A, D’Ottavio A, Leonetti F, Longobardi M (2001) Differentiability for bounded minimizers of some anisotropic integrals. J Math Anal Appl 253:640–650
Carlier G, Jimenez C, Santambrogio F (2008) Optimal transportation with traffic congestion and Wardrop equilibria. SIAM J Control Optim 47:1330–1350
Carstensen C, Müller S (2002) Local stress regularity in scalar nonconvex variational problems. SIAM J Math Anal 34:495–509
Correa JR, Stier-Moses NE (2011) Wardrop equilibria. In: Wiley encyclopedia of operations research and management science
Dacorogna B, Moser J (1990) On a partial differential equation involving the Jacobian determinant. Ann Inst Henri Poincaré, Anal Non Linéaire 7:1–26
De Pascale L, Pratelli A (2002) Regularity properties for Monge transport density and for solutions of some shape optimization problem. Calc Var Partial Differ Equ 14:249–274
De Pascale L, Pratelli A (2004) Sharp summability for Monge transport density via interpolation. ESAIM Control Optim Calc Var 10:549–552
De Pascale L, Evans LC, Pratelli A (2004) Integral estimates for transport densities. Bull Lond Math Soc 36(36):383–395
DiPerna RJ, Lions P-L (1989) Ordinary differential equations, transport theory and Sobolev spaces. Invent Math 98:511–547
Ekeland I, Temam R (1999) Convex analysis and variational problems. Classics in applied mathematics, vol 28. SIAM, Philadelphia
Knees D (2008) Global stress regularity of convex and some nonconvex variational problems. Ann Mat Pura Appl (4) 187:157–184
Lasry J-M, Lions P-L (2007) Mean-field games. Jpn J Math 2:229–260
Lindqvist P (2006) Notes on the p-Laplace equation. Report, University of Jyväskylä Department of Mathematics and Statistics, 102. University of Jyväskylä, Jyväskylä (2006). Available at http://www.math.ntnu.no/~lqvist/
Moser J (1965) On the volume elements on a manifold. Trans Am Math Soc 120:286–294
Nirenberg L (1955) Remarks on strongly elliptic partial differential equations. Commun Pure Appl Math 8:649–675
Santambrogio F (2009) Absolute continuity and summability of transport densities: simpler proofs and new estimates. Calc Var Partial Differ Equ 36:343–354
Wardrop JG (1952) Some theoretical aspects of road traffic research. Proc, Inst Civ Eng 2:325–378
Acknowledgement
We thank Filippo Santambrogio for some useful discussions. This work has been supported by the ANR through the projects ANR-09-JCJC-0096-01 EVAMEF and ANR-07-BLAN-0235 OTARIE, as well as by the ERC Advanced Grant no. 226234.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Brasco, L., Carlier, G. Congested Traffic Equilibria and Degenerate Anisotropic PDEs. Dyn Games Appl 3, 508–522 (2013). https://doi.org/10.1007/s13235-013-0081-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13235-013-0081-z