Science China Mathematics

, Volume 62, Issue 11, pp 2057–2072 | Cite as

Global regularity of optimal mappings in non-convex domains

  • Shibing Chen
  • Jiakun Liu
  • Xu-Jia WangEmail author


In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument, using a recent global regularity of optimal transportation in convex domains by the authors.


Monge-Ampère equation optimal transportation global regularity 


35J96 35J25 35B65 



This work was supported by Australian Research Council (Grant Nos. FL130100118 and DP170100929).


  1. 1.
    Bakelman I. Convex analysis and nonlinear geometric elliptic equations. Berlin: Springer-Verlag, 1994CrossRefGoogle Scholar
  2. 2.
    Brenier Y. Polar factorization and monotone rearrangement of vector-valued functions. Comm Pure Appl Math, 1991, 44: 375–417MathSciNetCrossRefGoogle Scholar
  3. 3.
    Caffarelli L. Interior W 2,p estimates for solutions of Monge-Ampere equations. Ann of Math (2), 1990, 131: 135–150MathSciNetCrossRefGoogle Scholar
  4. 4.
    Caffarelli L. The regularity of mappings with a convex potential. J Amer Math Soc, 1992, 5: 99–104MathSciNetCrossRefGoogle Scholar
  5. 5.
    Caffarelli L. Allocation maps with general cost functions. In: Partial Differential Equations and Applications. Lecture Notes in Pure and Applied Mathematics, 177. New York: Dekker, 1996, 29–35zbMATHGoogle Scholar
  6. 6.
    Caffarelli L. Boundary regularity of maps with convex potentials, II. Ann of Math (2), 1996, 144: 453–496MathSciNetCrossRefGoogle Scholar
  7. 7.
    Caffarelli L, Gonzáles M, Nguyen T. A perturbation argument for a Monge-Ampere equation with periodic data. Arch Ration Mech Anal, 2014, 212: 359–414MathSciNetCrossRefGoogle Scholar
  8. 8.
    Caffarelli L, McCann R. Free boundaries in optimal transport and Monge-Amere obstacle problems. Ann of Math (2), 2010, 171: 673–730MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen S. Boundary C 1,α regularity of an optimal transport problem with cost close tox. SIAM J Math Anal, 2015, 47: 2689–2698MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen S. Regularity of free boundaries in optimal transportation. PreprintGoogle Scholar
  11. 11.
    Chen S, Figalli A. Boundary ε-regularity in optimal transportation. Adv Math, 2015, 273: 540–567MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chen S, Figalli A. Partial W 2,p regularity for optimal transport maps. J Funct Anal, 2017, 272: 4588–4605MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chen S, Liu J, Wang X-J. Global regularity for the Monge-Ampere equation with natural boundary condition. ArXiv: 1802.07518Google Scholar
  14. 14.
    Chen S, Liu J, Wang X-J. Boundary regularity for the second boundary-value problem of Monge-Ampere equations in dimension two. ArXiv:1806.09482Google Scholar
  15. 15.
    De Philippis G, Figalli A. Partial regularity for optimal transport maps. Publ Math de l’IHES, 2015, 121: 81–112MathSciNetCrossRefGoogle Scholar
  16. 16.
    De Philippis G, Figalli A, Savin O. A note on interior W 2,1+ε estimates for the Monge-Ampere equation. Math Ann, 2013, 357: 11–22MathSciNetCrossRefGoogle Scholar
  17. 17.
    Delanoe Ph. Classical solvability in dimension two of the second boundary value problem associated with the Monge-Ampere operator. Ann Inst Henri Poincaré Anal Non Linéaire, 1991, 8: 443–457MathSciNetCrossRefGoogle Scholar
  18. 18.
    Evans L. Partial differential equations and Monge-Kantorovich mass transfer. In: Current Development in Mathematics. Boston: International Press, 1999, 65–126zbMATHGoogle Scholar
  19. 19.
    Figalli A. The Monge-Ampere Equation and its Applications. Zürich Lectures in Advanced Mathematics. Zürich: Euro Math Soc, 2017CrossRefGoogle Scholar
  20. 20.
    Gilbarg D, Trudinger N. Elliptic Partial Differential Equations of Second Order. Berlin: Springer-Verlag, 1983CrossRefGoogle Scholar
  21. 21.
    Gutierrez C. The Monge-Ampere Equation. Progress in Nonlinear Differential Equations and their Applications, 44. Boston: Birkhäuser, 2001CrossRefGoogle Scholar
  22. 22.
    Jian H Y, Wang X-J. Continuity estimates for the Monge-Ampere equation. SIAM J Math Anal, 2007, 39: 608–626MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kitagawa J, McCann R. Free discontinuities in optimal transport. Arch Ration Mech Anal, 2019, in presszbMATHGoogle Scholar
  24. 24.
    Lions P-L, Trudinger N, Urbas J. The Neumann problem for equations of Monge-Ampere type. Comm Pure Appl Math, 1986, 39: 539–563MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ma X N, Trudinger N, Wang X-J. Regularity of potential functions of the optimal transportation problem. Arch Ration Mech Anal, 2005, 177: 151–183MathSciNetCrossRefGoogle Scholar
  26. 26.
    Monge G. Memoire sur la Theorie des Déblais et des Remblais. In: Historie de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la M eme année, 1781, 666–704Google Scholar
  27. 27.
    Pogorelov A. Monge-Ampere Equations of Elliptic Type. Groningen: Noordhoff, 1964zbMATHGoogle Scholar
  28. 28.
    Savin O. Pointwise C 2,α estimates at the boundary for the Monge-Ampere equation. J Amer Math Soc, 2013, 26: 63–99MathSciNetCrossRefGoogle Scholar
  29. 29.
    Savin O. Global W 2,p estimates for the Monge-Ampere equations. Proc Amer Math Soc, 2013, 141: 3573–3578MathSciNetCrossRefGoogle Scholar
  30. 30.
    Trudinger N, Wang X-J. Boundary regularity of the Monge-Ampere and affne maximal surface equations. Ann of Math (2), 2008, 167: 993–1028MathSciNetCrossRefGoogle Scholar
  31. 31.
    Trudinger N, Wang X-J. The Monge-Ampere equation and its geometric applications. In: Handbook of Geometric Analysis, vol. 1. Adv Lect Math (ALM), vol. 7. Somerville: International Press, 2008, 467–524zbMATHGoogle Scholar
  32. 32.
    Urbas J. On the second boundary value problem of Monge-Ampère type. J Reine Angew Math, 1997, 487: 115–124MathSciNetzbMATHGoogle Scholar
  33. 33.
    Villani C. Optimal Transport. Old and New. Grundlehren Math Wiss, vol. 338. Berlin: Springer-Verlag, 2006zbMATHGoogle Scholar
  34. 34.
    Wang X-J, Wu Y T. A new proof for the regularity of Monge-Ampère type equations. J Differential Geom, 2019, in pressGoogle Scholar
  35. 35.
    Wolfson J. Minimal Lagrangian diffeomorphisms and the Monge-Ampère equation. J Differential Geom, 1997, 46: 335–373MathSciNetCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiChina
  2. 2.School of Mathematics and Applied StatisticsUniversity of WollongongWollongongAustralia
  3. 3.Centre for Mathematics and Its ApplicationsThe Australian National UniversityCanberraAustralia

Personalised recommendations