Skip to main content
SpringerLink
Log in

A description of transport cost for signed measures

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

In this paper, we develop the analysis started in a paper by Ambrosio, Mainini, and Serfaty about the extension of the optimal transport framework to the space of real measures. The main motivation comes from the study of nonpositive solutions to some evolution PDEs. Although a canonical optimal transport distance does not seem to be available, we may describe the cost for transporting signed measures in various ways and with interesting properties. Bibliography: 22 titles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. F. J. Almgren, J. Taylor, and L. Wang, “Curvature-driven flows: a variational approach,” SIAM J. Control Optim., 31, No. 2, 387–438 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  2. L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Spaces of Probability Measures, Birkhäuser Verlag, Basel (2005).

    MATH  Google Scholar 

  3. L. Ambrosio, E. Mainini, and S. Serfaty, “Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices,” Ann. Inst. H. Poincaré Anal. Non Linéaire, 28, No. 2, 217–246 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  4. L. Ambrosio and S. Serfaty, “A gradient flow approach to an evolution problem arising in superconductivity,” Comm. Pure Appl. Math., 61, No. 11, 1495–1539 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  5. V. I. Bogachev, Measure Theory, 2 volumes, Springer-Verlag, Berlin (2007).

    Book  MATH  Google Scholar 

  6. Y. Brenier, “Polar factorization and monotone rearrangement of vector-valued functions,” Comm. Pure Appl. Math., 44, 375–417 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  7. L. A. Caffarelli and R. J. McCann, “Free boundaries in optimal transport and Monge–Ampère obstacle problems,” Ann. Math. (2), 171, No. 2, 673–730 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  8. J. A. Carrillo, R. J. McCann, and C. Villani, “Contractions in the 2-Wasserstein length space and thermalization of granular media,” Arch. Ration. Mech. Anal., 179, No. 2, 217–263 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  9. J. S. Chapman, J. Rubinstein, and M. Schatzman, “A mean-field model for superconducting vortices,” European J. Appl. Math., 7, No. 2, 97–111 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  10. E. De Giorgi, “New problems on minimizing movements,” in: C. Baiocchi, and J. L. Lions (eds.), Boundary Value Problems for PDE and Applications, Masson, Paris (1993), pp. 81–98.

    Google Scholar 

  11. W. E, “Dynamics of vortex-liquies in Ginzburg–Landau theories with applications to superconductivity,” Phys. Rev. B, 50, No. 3, 1126–1135 (1994).

    Article  MathSciNet  Google Scholar 

  12. A. Figalli, “The optimal partial transport problem,” Arch. Ration. Mech. Anal., 195, No. 2, 533–560 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  13. W. Gangbo and R. McCann, “The geometry of optimal transport,” Acta Math., 177, 113–161 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  14. R. Jordan, D. Kinderlehrer, and F. Otto, “The variational formulation of the Fokker–Planck equation,” SIAM J. Math. Anal., 29, 1–17 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  15. L. V. Kantorovich, “On the transfer of masses,” Dokl. Akad. Nauk SSSR, 37, 227–229 (1942).

    Google Scholar 

  16. L. V. Kantorovich, “On a problem of Monge,” Uspekhi Mat. Nauk, 3, 225–226 (1948).

    Google Scholar 

  17. E. Mainini, “A global uniqueness result for an evolution problem arising in superconductivity,” Boll. Unione Mat. Ital. (9), 2, No. 2, 509–528 (2009).

    MathSciNet  MATH  Google Scholar 

  18. E. Mainini, “Well-posedness for a mean field model of Ginzburg–Landau vortices with opposite degrees,” NoDEA Nonlinear Differential Equations Appl., in press.

  19. F. Otto, “The geometry of dissipative evolution equations: the porous-medium equation,” Comm. Partial Differential Equations, 26, 101–174 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  20. C. Villani, Topics in Optimal Transportation, Amer. Math. Soc., Providence, Rhode Island (2003).

    MATH  Google Scholar 

  21. C. Villani, Optimal Transport, Old and New, Springer-Verlag, Berlin (2008).

    Google Scholar 

  22. G. Wolanski, “Limit theorems for optimal mass transportation,” Calc. Var. Partial Differential Equations, in press.

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Pavia, Italy

    E. Mainini

Authors
  1. E. Mainini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Mainini.

Additional information

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 390, pp. 147–181.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mainini, E. A description of transport cost for signed measures. J Math Sci 181, 837–855 (2012). https://doi.org/10.1007/s10958-012-0718-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-012-0718-2

Keywords

Search

Navigation