Dynamics of optimal partial transport


Optimal partial transport, which was initially studied by Caffarelli and McCann (Ann Math (2) 171(2):673–730, 2010), is a variant of optimal transport theory, where only a portion of mass is to be transported in an efficient way. Free boundaries naturally arise as the boundary of the region where the actual transport occurs. This paper considers the evolution dynamics of the free boundaries in terms of the change of m, the allowed amount of transported mass or the change of \(\lambda \), the transportation cost cap, i.e. the allowed maximum cost for a unit mass to be transported. Focusing on the quadratic cost function, we show Hölder and Lipschitz estimates on the speed of the free boundary motion in terms of m and \(\lambda \), respectively. It is also shown that the parameter m is a Lipschitz function of \(\lambda \), which previously was known only to be a continuous increasing function (Caffarelli and McCann Ann Math (2) 171(2):673–730, 2010).

This is a preview of subscription content, access via your institution.


  1. 1.

    Agueh, M., Carlier, G.: Barycenters in the Wasserstein space. SIAM J. Math. Anal. 43(2), 904–924 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Caffarelli, L.A.: Interior \(W^{2,p}\) estimates for solutions of the Monge-Ampère equation. Ann. Math. (2) 131(1), 135–150 (1990)

  3. 3.

    Caffarelli, L.A., McCann, R.J.: Free boundaries in optimal transport and Monge-Ampère obstacle problems. Ann. Math. (2) 171(2), 673–730 (2010)

  4. 4.

    Chen, S., Indrei, E.: On the regularity of the free boundary in the optimal partial transport problem for general cost functions. J. Differ. Equ. 258(7), 2618–2632 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Figalli, A.: A note on the regularity of the free boundaries in the optimal partial transport problem. Rend. Circ. Mat. Palermo (2) 58(2), 283–286 (2009)

  6. 6.

    Figalli, A.: The optimal partial transport problem. Arch. Ration. Mech. Anal. 195(2), 533–560 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Gangbo, W., McCann, R.J.: The geometry of optimal transportation. Acta Math. 177(2), 113–161 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin (2001, Reprint of the 1998 edition)

  9. 9.

    Indrei, E.: Free boundary regularity in the optimal partial transport problem. J. Funct. Anal. 264(11), 2497–2528 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Indrei, E., Nurbekyan, L.: Regularity of shadows and the geometry of the singular set associated to a monge-ampere equation. Commun. Anal. Geom. (to appear)

  11. 11.

    Kantorovich, L.V.: On mass transportation. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 312(Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 11), 11–14 (2004)

  12. 12.

    Kim, Y.-H., Pass, B.: Wasserstein barycenters over riemannian manfolds (2014). arXiv:1412.7726

  13. 13.

    Kitagawa, J., Pass, B.: The multi-marginal optimal partial transport problem (2014). arXiv:1401.7255

  14. 14.

    Loeper, G.: On the regularity of solutions of optimal transportation problems. Acta Math. 202(2), 241–283 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Ma, X.-N., Trudinger, N.S., Wang, X.-J.: Regularity of potential functions of the optimal transportation problem. Arch. Ration. Mech. Anal. 177(2), 151–183 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    McCann, R.J.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80(2), 309–323 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Monge, G.: Mémoire sur la théorie des déblais et de remblais. In: Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même Année, pp. 666–704 (1781)

  18. 18.

    Trudinger, N.S., Wang, X.-J.: On the second boundary value problem for Monge-Ampère type equations and optimal transportation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8(1), 143–174 (2009)

  19. 19.

    De Philippis, G., Mészáros, A.R., Santambrogio, F., Velichkov, B.: Bv estimates in optimal transportation and applications. http://cvgmt.sns.it/paper/2559/

  20. 20.

    Villani, C.: Topics in optimal transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003)

  21. 21.

    Villani, C.: Optimal transport: old and new, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338. Springer-Verlag, Berlin (2009)

Download references


We thank Inwon Kim for helpful discussions and interest in this work. We also thank the anonymous referee who suggested several improvements.

Author information



Corresponding author

Correspondence to Gonzalo Dávila.

Additional information

This research is partially supported by Natural Sciences and Engineering Research Council of Canada Discovery Grants 371642-09 and 2014-05448 as well as the Alfred P. Sloan Research Fellowship 2012–2016. Part of this research has been done while Y.-H.K. was visiting Korea Advanced Institute of Science and Technology (KAIST), the Mathematical Sciences Research Institute (MSRI) for the thematic program “Optimal Transport: Geometry and Dynamics”, and the Fields institute, Toronto for the thematic program on “Calculus of Variations”. ©2015 by the authors.

Communicated by N.Trudinger.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dávila, G., Kim, YH. Dynamics of optimal partial transport. Calc. Var. 55, 116 (2016). https://doi.org/10.1007/s00526-016-1052-1

Download citation

Mathematics Subject Classification

  • 46N10
  • 28A75
  • 35J96
  • 35R35