Skip to main content
SpringerLink
Log in

BV Estimates in Optimal Transportation and Applications

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

In this paper we study the BV regularity for solutions of certain variational problems in Optimal Transportation. We prove that the Wasserstein projection of a measure with BV density on the set of measures with density bounded by a given BV function f is of bounded variation as well and we also provide a precise estimate of its BV norm. Of particular interest is the case f = 1, corresponding to a projection onto a set of densities with an L ∞ bound, where we prove that the total variation decreases by projection. This estimate and, in particular, its iterations have a natural application to some evolutionary PDEs as, for example, the ones describing a crowd motion. In fact, as an application of our results, we obtain BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation techniques. We also establish some properties of the Wasserstein projection which are interesting in their own right, and allow, for instance, for the proof of the uniqueness of such a projection in a very general framework.

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

Similar content being viewed by others

References

  1. Ambrosio L.: Movimenti minimizzanti. Rend. Accad. Naz. Sci. XL Mem. Mat. Sci. Fis. Nat. 113, 191–246 (1995)

    MathSciNet  Google Scholar 

  2. Ambrosio, L., Gigli, N., Savaré, G.: Gradient flows in metric spaces and in the space of probability measures, Lectures in Math., ETH Zürich, (2005)

  3. Bouchitté G., Buttazzo G.: New lower semicontinuity results for nonconvex functionals defined on measures. Nonlinear Anal. 15(7), 679–692 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. Buttazzo, G.: Semicontinuity, Relaxation, and Integral Representation in the Calculus of Variations. Longman Scientific & Technical, Essex, 1989

  5. Buttazzo, G., Santambrogio, F.: A model for the optimal planning of an urban area. SIAM J. Math. Anal. 37(2), 514–530 (2005)

  6. Carlen, E., Craig, K.: Contraction of the proximal map and generalized convexity of the Moreau–Yosida regularization in the 2-Wasserstein metric. Math. Mech. Complex Syst. 1(1), 33–65 (2013)

  7. Carlier, G., Santambrogio, F.: A variational model for urban planning with traffic congestion. ESAIM Control Optim. Calc. Var. 11(4), 595–613 (2005)

  8. Caffarelli L. A.: The regularity of mappings with a convex potential. J. Am. Math. Soc. 5(1), 99–104 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Caffarelli L. A. (1992) Boundary regularity of maps with convex potentials. Commun. Pure Appl. Math. 45(9): 1141–1151

  10. Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control. Birkhäuser, Basel, 2004

  11. Dal Maso, G.: An Introduction to Γ-Convergence. Birkhäuser, Basel, 1992

  12. Figalli, A.: A note on the regularity of the free boundaries in the optimal partial transport problem. Rend. Circ. Mat. Palermo 58, 283–286 (2009)

  13. Figalli, A.: The optimal partial transport problem. Arch. Rat. Mech. Anal. 195, 533–560 (2010)

  14. Gigli, N.: On the inverse implication of Brenier–McCann theorems and the structure of \({(\mathcal{P}_{2}(M),W_2)}\). Meth. Appl. Anal. 18(2), 127–158 (2011)

  15. Lellmann, J., Lorenz, D.A., Schönlieb, C., Valkonen, T.: Imaging with Kantorovich–Rubinstein discrepancy. SIAM J. Imaging Sci. 7(4), 2833–2859 (2014)

  16. Maury, B., Roudneff-Chupin, A., Santambrogio, F.: A macroscopic crowd motion model of gradient flow type. Math. Models Methods Appl. Sci. 20(10), 1787–1821 (2010)

  17. Maury, B., Roudneff-Chupin, A., Santambrogio F.: Congestion-driven dendritic growth. Discr. Cont. Dyn. Syst. 34(4), 1575–1604 (2014)

  18. McCann, R. J.: A convexity principle for interacting gases. Adv. Math. 128(1), 153–159 (1997)

  19. Milakis, E.: On the regularity of optimal sets in mass transfer problems. Commun. Partial Diff. Equ. 31(4–6), 817–826 (2006)

  20. Mészáros, A.R., Santambrogio, F.: A diffusive model for macroscopic crowd motion with density constraints (preprint). http://cvgmt.sns.it/paper/2644/

  21. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. PDE 26(1-2) 101–174 (2001)

  22. Roudneff-Chupin, A.: Modélisation macroscopique de mouvements de foule, PhD Thesis, Université Paris-Sud. http://www.math.u-psud.fr/~roudneff/Images/these_roudneff (2011)

  23. Santambrogio F.: Transport and concentration problems with interaction effects. J. Glob. Optim. 38(1), 129–141 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Santambrogio, F.: Optimal Transport for Applied Mathematicians. Birkäuser, New York, 2015

  25. Villani, Cédric: Optimal Transport Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer, Berlin, 2009

  26. Vázquez, J. L.: The Porous Medium Equation. Mathematical Theory. The Clarendon Press, Oxford University Press, Oxford, 2007

Download references

Author information

Authors and Affiliations

  1. UMPA, CNRS and École Normale Supérieure de Lyon, 46, Allée d’Italie, 69364, Lyon Cedex 07, France

    Guido De Philippis

  2. Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, 91405, Orsay Cedex, France

    Alpár Richárd Mészáros & Filippo Santambrogio

  3. Laboratoire LJK, Université J. Fourier, 51, rue des Mathématiques, 38041, Grenoble Cedex 9, France

    Bozhidar Velichkov

Authors
  1. Guido De Philippis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alpár Richárd Mészáros
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Filippo Santambrogio
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bozhidar Velichkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Filippo Santambrogio.

Additional information

Communicated by G. Dal Maso

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

De Philippis, G., Mészáros, A.R., Santambrogio, F. et al. BV Estimates in Optimal Transportation and Applications. Arch Rational Mech Anal 219, 829–860 (2016). https://doi.org/10.1007/s00205-015-0909-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-015-0909-3

Keywords

Search

Navigation