Advertisement

Archive for Rational Mechanics and Analysis

, Volume 215, Issue 3, pp 867–905 | Cite as

On Asymptotic Behaviour and W 2, p Regularity of Potentials in Optimal Transportation

  • Jiakun Liu
  • Neil S. Trudinger
  • Xu-Jia Wang
Article

Abstract

In this paper we study local properties of cost and potential functions in optimal transportation. We prove that in a proper normalization process, the cost function is uniformly smooth and converges locally smoothly to a quadratic cost x · y, while the potential function converges to a quadratic function. As applications we obtain the interior W 2, p estimates and sharp C 1, α estimates for the potentials, which satisfy a Monge–Ampère type equation. The W 2, p estimate was previously proved by Caffarelli for the quadratic transport cost and the associated standard Monge–Ampère equation.

Keywords

Cost Function Asymptotic Behaviour Potential Function Comparison Principle Good Shape 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brenier Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44, 375–417 (1991)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Caffarelli, L.A.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. (2) 130, 189–213 (1989)Google Scholar
  3. 3.
    Caffarelli, L.A.: A localization property of viscosity solutions to the Monge–Ampère equation and their strict convexity. Ann. Math. (2) 131, 129–134 (1990)Google Scholar
  4. 4.
    Caffarelli L.A.: Interior W 2, p estimates for solutions of Monge–Ampère equations. Ann. Math. 131, 135–150 (1990)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Caffarelli, L.A.: Allocation maps with general cost functions. In: Partial Differential Equations and Applications. Lecture Notes in Pure and Applied Mathematics, vol. 177, pp. 29–35. Dekker, New York, 1996Google Scholar
  6. 6.
    Caffarelli, L.A.: Nonlinear elliptic theory and the Monge–Ampère equation. In: Proceedings of the International Congress of Mathematics, vol.1, pp. 179–187, 2002Google Scholar
  7. 7.
    Chou K.S., Wang X.-J.: Entire solutions of the Monge–Ampère equation. Commun. Pure Appl. Math. 49, 529–539 (1996)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    De Philippis, G., Figalli, A.: Sobolev regularity for Monge–Ampère type equations. SIAM J. Math. Anal. 45, 1812–1824 (2013)Google Scholar
  9. 9.
    De Philippis, G., Figalli, A.: W 2,1 regularity for solutions of the Monge–Ampère equation. Invent. Math. 192, 55–69 (2013)Google Scholar
  10. 10.
    De Philippis, G., Figalli, A., Savin, O.: A note on interior W 2, 1+ɛ estimates for the Monge–Ampère equation. Math. Ann. 357, 11–22 (2013)Google Scholar
  11. 11.
    Figalli A., Kim Y.-M., McCann R.: Hölder continuity and injectivity of optimal maps. Arch. Ration. Mech. Anal. 209, 747–795 (2013)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Figalli A., Rifford L., Villani C.: On the Ma–Trudinger–Wang curvature on surfaces. Calc. Var. Partial Differ. Equ. 39, 307–332 (2010)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Gangbo W., McCann R.J.: The geometry of optimal transportation. Acta Math. 177, 113–161 (1996)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Gilbarg D., Trudinger N. S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)CrossRefMATHGoogle Scholar
  15. 15.
    Gutiérrez, C.E.: The Monge–Ampère Equation. Progress in Nonlinear Differential Equations and their Applications, vol. 44. Birkhäuser Boston Inc., Boston, 2001Google Scholar
  16. 16.
    Jian H.Y., Wang X.-J.: Continuity estimates for the Monge–Ampère equation. SIAM J. Math. Anal. 39,–608626 (2007)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    John, F.: Extremum problems with inequalities as subsidiary conditions. Studies and Essays Presented to R. Courant on his 60th Birthday, pp. 187–204. Interscience Publishers, New York, 1948Google Scholar
  18. 18.
    Kim Y.H., McCann R.: Continuity, curvature, and the general covariance of optimal transportation. J. Eur. Math. Soc. 12, 1009–1040 (2010)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Levin V.: Abstract cyclical monotonicity and Monge solutions for the general Monge–Kantorovich problem. Set Valued Anal. 7, 7–32 (1999)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Liu J.: Hölder regularity of optimal mappings in optimal transportation. Calc. Var. Partial Differ. Equ. 34, 435–451 (2009)CrossRefMATHGoogle Scholar
  21. 21.
    Liu J., Trudinger N. S.: On Pogorelov estimates for Monge–Ampere type equations. Discret. Contin. Dyn. Syst. 28, 1121–1135 (2010)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Liu, J., Trudinger, N.S., Wang, X.-J.: Interior C 2,α regularity for potential functions in optimal transportation. Commun. Partial Differ. Equ., 35, 165–184 (2010)Google Scholar
  23. 23.
    Loeper, G.: On the regularity of solutions of optimal transportation problems. Acta Math. 202, 241–283 (2009)Google Scholar
  24. 24.
    Ma, X.N., Trudinger, N.S., Wang, X.-J.: Regularity of potential functions of the optimal transportation problem. Arch. Ration. Mech. Anal., 177, 151–183 (2005)Google Scholar
  25. 25.
    Pogorelov, A. V.: The Minkowski Multidimensional Problem. Scripta Series in Mathematics. Wiley, New York, 1978Google Scholar
  26. 26.
    Schmidt, T.: W 2, 1+ɛ estimates for the Monge–Ampère equation. Adv. Math. 240, 672–689 (2013)Google Scholar
  27. 27.
    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, 2006Google Scholar
  28. 28.
    Trudinger, N.S., Wang, X.-J.: The Monge–Ampère equation and its geometric applications. Handbook of Geometric Analysis, vol. 1, pp. 467–524. Advanced Lectures in Mathematics (ALM), vol. 7, International Press, Somerville, 2008Google Scholar
  29. 29.
    Trudinger N.S., Wang X.-J.: On strict convexity and continuous differentiability of potential functions in optimal transportation. Arch. Ration. Mech. Anal. 192, 403–418 (2009)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    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, 143–174 (2009)Google Scholar
  31. 31.
    Urbas, J.: Mass Transfer Problems. Lecture Notes. University of Bonn, 1998Google Scholar
  32. 32.
    Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence, 2003Google Scholar
  33. 33.
    Villani, C.: Optimal Transport. Old and New. Grundlehren Der Mathematischen Wissenschaften, vol. 338. Springer, Berlin, 2009Google Scholar
  34. 34.
    Villani C.: Stability of a 4th-order curvature condition arising in optimal transport theory. J. Funct. Anal. 255, 2683–2708 (2008)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Wang X.-J.: On the design of a reflector antenna. Inverse Prob. 12, 351–375 (1996)ADSCrossRefMATHGoogle Scholar
  36. 36.
    Wang X.-J.: Some counterexamples to the regularity of Monge–Ampère equations. Proc. Am. Math. Soc. 123(3), 841–845 (1995)MATHGoogle Scholar
  37. 37.
    Wang X.-J.: Schauder estimate for elliptic and parabolic equations. Chin. Ann. Math., Series B, 27, 637–642 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute for Mathematics and its Applications, School of Mathematics and Applied StatisticsUniversity of WollongongWollongongAustralia
  2. 2.Centre for Mathematics and Its ApplicationsThe Australian National UniversityCanberraAustralia

Personalised recommendations