Skip to main content
Log in

Regularity of Potential Functions of the Optimal Transportation Problem

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

Abstract

The potential function of the optimal transportation problem satisfies a partial differential equation of Monge-Ampère type. In this paper we introduce the notion of a generalized solution, and prove the existence and uniqueness of generalized solutions of the problem. We also prove the solution is smooth under certain structural conditions on the cost function.

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

Access this article

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. Aleksandrov, A.D.: Existence and uniqueness of a convex surface with a given integral curvature. C.R. (Doklady) Acad. Sci.URSS (N.S.) 35, 131–134 (1942)

    Google Scholar 

  2. Ambrosio, L.: Optimal transport maps in Monge-Kantorovich problem. In: Proc. International Cong. Math. Beijing, Vol.3, 131–140 (2002)

  3. Bakelman, I.J.: Convex analysis and nonlinear geometric elliptic equations. Springer, Berlin, 1994

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

    Google Scholar 

  5. Brenier, Y.: Extended Monge-Kantorovich theory. CIME lecture notes, 2001

  6. Caffarelli, L.: Allocation maps with general cost functions, in Partial Differential Equations and Applications (P. Marcellini, G. Talenti, and E. Vesintini eds). Lecture Notes in Pure and Appl. Math. 177, 29–35 (1996)

    Google Scholar 

  7. Caffarelli, L.: The regularity of mappings with a convex potential. J. Amer. Math. Soc. 5, 99–104 (1992)

    Google Scholar 

  8. Caffarelli, L.: Boundary regularity of maps with convex potentials II. Ann. of Math. 144, 453–496 (1996)

    Google Scholar 

  9. Caffarelli, L., Feldman, M., McCann, R.J.: Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs. J. Amer. Math. Soc. 15, 1–26 (2002)

    Article  Google Scholar 

  10. Cheng, S.Y., Yau, S.T.: On the regularity of the Monge-Ampère equation det(∂2 u/∂ x i x j )=F(x,u). Comm. Pure Appl. Math. 30, 41–68 (1977)

    Google Scholar 

  11. Chou, K.S., Wang, X.J.: Minkowski problem for complete noncompact convex hypersurfaces. Topolo. Methods in Nonlinear Anal. 6, 151–162 (1995)

    Google Scholar 

  12. Delanoë, PH.: Classical solvability in dimension two of the second boundary value problem associated with the Monge-Ampère operator. Ann. Inst. Henri Poincaré, Analyse Non Linéaire 8, 443–457 (1991)

    Google Scholar 

  13. Evans, L.C.: Partial differential equations and Monge-Kantorovich mass transfer. In: Current development in mathematics. Int. Press, Boston, 65–126 1999

  14. Evans, L.C., Gangbo, W.: Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc., 653, 1999

  15. Gangbo, W., McCann, R.J.: Optimal maps in Monge’s mass transport problem. C.R. Acad. Sci. Paris, Series I, Math. 321, 1653–1658 (1995)

    Google Scholar 

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

    Google Scholar 

  17. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer, 1983

  18. Guan, P., Wang, X.J.: On a Monge-Ampère equation arising in geometric optics. J. Diff. Geom. 48, 205–223 (1998)

    Google Scholar 

  19. Hong, J.X.: Dirichlet problems for general Monge-Ampère equations. Math. Z. 209, 289–306 (1992)

    Google Scholar 

  20. Lions, P.L., Trudinger, N.S., Urbas, J.: The Neumann problem for equations of Monge-Ampère type. In: Proceedings of the Centre for Mathematical Analysis. Australian National University, 10, 135–140 (1985)

    Google Scholar 

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

    Article  Google Scholar 

  22. 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 Même année, 1781, 666–704

  23. Pogorelov, A.V.: Monge-Ampère equations of elliptic type. Noordhoff, Groningen, 1964

  24. Pogorelov, A.V.: The multidimensional Minkowski problem. Wiley, New York, 1978

  25. Rachev, S.T., Ruschendorff, L.: Mass transportation problems. Springer, Berlin, 1998

  26. Schulz, F.: Regularity theory for quasilinear elliptic systems and Monge-Ampère equations in two dimensions. Lecture Notes in Math. 1445, 1990

  27. Trudinger, N.S.: On the Dirichlet problem for Hessian equations. Acta Math. 175, 151–164 (1995)

    Google Scholar 

  28. Trudinger, N.S.: Lectures on nonlinear elliptic equations of second order. Lectures in Mathematical Sciences 9, Univ. Tokyo, 1995

  29. Trudinger. N.S., Wang, X.J.: On the Monge mass transfer problem. Calc. Var. PDE 13, 19–31 (2001)

    Google Scholar 

  30. Urbas, J.: On the second boundary value problem for equations of Monge-Ampère type. J. Reine Angew. Math. 487, 115–124 (1997)

    Google Scholar 

  31. Urbas, J.: Mass transfer problems. Lecture Notes, Univ. of Bonn, 1998

  32. Wang, X.J.: On the design of a reflector antenna. Inverse Problems 12, 351–375 (1996)

    Article  Google Scholar 

  33. Wang, X.J.: On the design of a reflector antenna II. Calc. Var. PDE 20, 329–341 (2004)

    Article  Google Scholar 

  34. Wang, X.J.: Oblique derivative problems for Monge-Ampère equations(Chinese). Chinese Ann. Math. Ser. A 13, 41–50 (1992)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Neil S. Trudinger.

Additional information

Communicated by L. Ambrosio

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ma, XN., Trudinger, N. & Wang, XJ. Regularity of Potential Functions of the Optimal Transportation Problem. Arch. Rational Mech. Anal. 177, 151–183 (2005). https://doi.org/10.1007/s00205-005-0362-9

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-005-0362-9

Keywords

Navigation