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Congested Traffic Equilibria and Degenerate Anisotropic PDEs

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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.

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Notes

  1. 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 −.

  2. Interestingly, the connection with the Monge–Kantorovich theory was realized much later by Robert McCann.

References

  1. Ambrosio L (2004) Transport equation and Cauchy problem for BV vector fields. Invent Math 158:227–260

    Article  MathSciNet  MATH  Google Scholar 

  2. Baillon J-B, Carlier G (2012) From discrete to continuous Wardrop equilibria. Netw Heterog Media 7:219–241

    Article  MathSciNet  MATH  Google Scholar 

  3. Beckmann MJ (1952) A continuous model of transportation. Econometrica 20:643–660

    Article  MathSciNet  MATH  Google Scholar 

  4. Beckmann M, McGuire C, Winsten C (1956) Studies in economics of transportation. Yale University Press, New Haven

    Google Scholar 

  5. Belloni M, Kawohl B (2004) The pseudo p-Laplace eigenvalue problem and viscosity solutions as p→∞. ESAIM Control Optim Calc Var 10:28–52

    Article  MathSciNet  MATH  Google Scholar 

  6. 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/

  7. Brasco L, Carlier G, Santambrogio F (2010) Congested traffic dynamics, weak flows and very degenerate elliptic equations. J Math Pures Appl 93:652–671

    MathSciNet  MATH  Google Scholar 

  8. 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

    Article  MathSciNet  MATH  Google Scholar 

  9. Carlier G, Jimenez C, Santambrogio F (2008) Optimal transportation with traffic congestion and Wardrop equilibria. SIAM J Control Optim 47:1330–1350

    Article  MathSciNet  MATH  Google Scholar 

  10. Carstensen C, Müller S (2002) Local stress regularity in scalar nonconvex variational problems. SIAM J Math Anal 34:495–509

    Article  MathSciNet  MATH  Google Scholar 

  11. Correa JR, Stier-Moses NE (2011) Wardrop equilibria. In: Wiley encyclopedia of operations research and management science

    Google Scholar 

  12. Dacorogna B, Moser J (1990) On a partial differential equation involving the Jacobian determinant. Ann Inst Henri Poincaré, Anal Non Linéaire 7:1–26

    MathSciNet  MATH  Google Scholar 

  13. 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

    Article  MATH  Google Scholar 

  14. De Pascale L, Pratelli A (2004) Sharp summability for Monge transport density via interpolation. ESAIM Control Optim Calc Var 10:549–552

    Article  MathSciNet  MATH  Google Scholar 

  15. De Pascale L, Evans LC, Pratelli A (2004) Integral estimates for transport densities. Bull Lond Math Soc 36(36):383–395

    Article  MATH  Google Scholar 

  16. DiPerna RJ, Lions P-L (1989) Ordinary differential equations, transport theory and Sobolev spaces. Invent Math 98:511–547

    Article  MathSciNet  MATH  Google Scholar 

  17. Ekeland I, Temam R (1999) Convex analysis and variational problems. Classics in applied mathematics, vol 28. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  18. Knees D (2008) Global stress regularity of convex and some nonconvex variational problems. Ann Mat Pura Appl (4) 187:157–184

    Article  MathSciNet  MATH  Google Scholar 

  19. Lasry J-M, Lions P-L (2007) Mean-field games. Jpn J Math 2:229–260

    Article  MathSciNet  MATH  Google Scholar 

  20. 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/

  21. Moser J (1965) On the volume elements on a manifold. Trans Am Math Soc 120:286–294

    Article  MATH  Google Scholar 

  22. Nirenberg L (1955) Remarks on strongly elliptic partial differential equations. Commun Pure Appl Math 8:649–675

    Article  MathSciNet  MATH  Google Scholar 

  23. Santambrogio F (2009) Absolute continuity and summability of transport densities: simpler proofs and new estimates. Calc Var Partial Differ Equ 36:343–354

    Article  MathSciNet  MATH  Google Scholar 

  24. Wardrop JG (1952) Some theoretical aspects of road traffic research. Proc, Inst Civ Eng 2:325–378

    Google Scholar 

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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.

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Authors and Affiliations

  1. LATP, Aix-Marseille Université, 13453, Marseille, France

    L. Brasco

  2. CEREMADE, UMR CNRS 7534, Université Paris IX Dauphine, 75775, Paris, France

    G. Carlier

Authors
  1. L. Brasco
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  2. G. Carlier
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Correspondence to G. Carlier.

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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

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