Advertisement

Free Discontinuities in Optimal Transport

  • Jun KitagawaEmail author
  • Robert McCann
Article

Abstract

We prove a nonsmooth implicit function theorem applicable to the zero set of the difference of convex functions. This theorem is explicit and global: it gives a formula representing this zero set as a difference of convex functions which holds throughout the entire domain of the original functions. As applications, we prove results on the stability of singularities of envelopes of semi-convex functions, and solutions to optimal transport problems under appropriate perturbations, along with global structure theorems on certain discontinuities arising in optimal transport maps for the bilinear cost \({c(x, \bar{x}):=-\langle {x}, {\bar{x}}\rangle}\) for \({x,\bar{x} \in {\bf R}^n}\). For targets whose components satisfy additional convexity, separation, multiplicity, and affine independence assumptions, we show that these discontinuities occur on submanifolds of the appropriate codimension which are parameterized locally as differences of convex functions (DC, hence \({C^2}\) rectifiable), and—depending on the precise assumptions—\({C^{1,\alpha}}\) smooth. Under these hypotheses, any \({n+1}\) affinely independent components of the target measure select at most one point from the source measure where the transport divides between all \({n+1}\) specified target components.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflicts of interest.

References

  1. 1.
    Alberti, G.: On the structure of singular sets of convex functions. Calc. Var. Part. Differ. Equ. 2(1), 17–27 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Bertrand, J.: DC calculus. Math. Z. 288(3–4), 1037–1080 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Butler, G.J., Timourian, J.G., Viger, C.: The rank theorem for locally Lipschitz continuous functions. Can. Math. Bull. 31(2), 217–226 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Caffarelli, L.A.: The regularity of mappings with a convex potential. J. Am. Math. Soc. 5(1), 99–104 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Caffarelli, L.A., McCann, R.J.: Free boundaries in optimal transport and Monge–Ampère obstacle problems. Ann. Math. (2) 171(2), 673–730 (2010)Google Scholar
  7. 7.
    Chen, S.: Regularity of free boundary in optimal transportation. (Preprint)Google Scholar
  8. 8.
    Chodosh, O., Jain, V., Lindsey, M., Panchev, L., Rubinstein, Y.A.: On discontinuity of planar optimal transport maps. J. Topol. Anal. 7(2), 239–260 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Clarke, F.H.: On the inverse function theorem. Pac. J. Math. 64(1), 97–102 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Clarke, F.H.: Optimization and Nonsmooth Analysis, Volume 5 of Classics in Applied Mathematics, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1990)Google Scholar
  11. 11.
    De Philippis, G., Figalli, A.: Partial regularity for optimal transport maps. Publ. Math. Inst. Hautes Études Sci. 121, 81–112 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Figalli, A.: Regularity properties of optimal maps between nonconvex domains in the plane. Commun. Part. Differ. Equ. 35(3), 465–479 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Figalli, A., Kim, Y.-H.: Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete Contin. Dyn. Syst. 28(2), 559–565 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gigli, N.: On the inverse implication of Brenier-McCann theorems and the structure of \(({\cal{P}}_2(M), W_2)\). Methods Appl. Anal. 18(2), 127–158 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Goldman, M., Otto, F.: A variational proof for partial regularity of optimal transportation maps. Preprint at arXiv:1704.05339v1
  16. 16.
    Jian, H.-Y., Wang, X.-J.: Continuity estimates for the Monge-Ampère equation. SIAM J. Math. Anal. 39(2), 608–626 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Liu, J., Trudinger, N.S., Wang, X.-J.: Interior \(C^{2,\alpha }\) regularity for potential functions in optimal transportation. Commun. Part. Differ. Equ. 35(1), 165–184 (2010)CrossRefzbMATHGoogle Scholar
  18. 18.
    Loeper, G.: On the regularity of solutions of optimal transportation problems. Acta Math. 202(2), 241–283 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    McCann, R.J.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80(2), 309–323 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Perelman, G.: DC structure on Alexandrov space (preliminary version). https://www.math.psu.edu/petrunin/papers/akp-papers/perelman-DC.pdf
  22. 22.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  23. 23.
    Trudinger, N.S.: Recent developments in elliptic partial differential equations of Monge–Ampère type. In: International Congress of Mathematicians, vol. III, pp. 291–301. European Mathematical Society, Zürich (2006)Google Scholar
  24. 24.
    Trudinger, N.S., Wang, X.-J.: On the second boundary value problem for Monge–Ampère type equations and optimal transportation. Ann. Sci. Norm. Super. Pisa Class. Sci. 8(1), 143–174 (2009)Google Scholar
  25. 25.
    Veselý, L., Zajíček, L.: Delta-convex mappings between Banach spaces and applications. Diss. Math. (Rozprawy Mat.) 289, 52 (1989)Google Scholar
  26. 26.
    Villani, C.: Topics in Optimal Transportation, Volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2003)Google Scholar
  27. 27.
    Villani, C.: Optimal Transport: Old and New, Volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2009)Google Scholar
  28. 28.
    Wang, X.-J.: Remarks on the Regularity of Monge–Ampère, pp. 257–263. Academic Press, Beijing (1992)Google Scholar
  29. 29.
    Warga, J.: An implicit function theorem without differentiability. Proc. Am. Math. Soc. 69(1), 65–69 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Zajíček, L.: On the differentiation of convex functions in finite and infinite dimensional spaces. Czechoslov. Math. J. 29(104)(3), 340–348 (1979)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada

Personalised recommendations