Advertisement

Regularity of the Monge–Ampère equation in Besov’s spaces

  • Alexander V. Kolesnikov
  • Sergey Yu. TikhonovEmail author
Article

Abstract

Let \(\mu = e^{-V} \ dx\) be a probability measure and \(T = \nabla \Phi \) be the optimal transportation mapping pushing forward \(\mu \) onto a log-concave compactly supported measure \(\nu = e^{-W} \ dx\). In this paper, we introduce a new approach to the regularity problem for the corresponding Monge–Ampère equation \(e^{-V} = \det D^2 \Phi \cdot e^{-W(\nabla \Phi )}\) in the Besov spaces \(W^{\gamma ,1}_{loc}\). We prove that \(D^2 \Phi \in W^{\gamma ,1}_{loc}\) provided \(e^{-V}\) belongs to a proper Besov class and \(W\) is convex. In particular, \(D^2 \Phi \in L^p_{loc}\) for some \(p>1\). Our proof does not rely on the previously known regularity results.

Mathematics Subject Classification (2000)

Primary 35J60 35B65 Secondary 46E35 

References

  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Wasserstein Spaces of Probability Measures. Birkhäuser, Brazil (2008)Google Scholar
  3. 3.
    Bakelman, I.J.: Convex Analysis and Nonlinear Geometric Elliptic Equations. Springer, Berlin (1994)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bogachev, V.I., Kolesnikov, A.V.: Sobolev regularity for the Monge–Ampere equation in the Wiener space. arXiv: 1110.1822 (to appear in Kyoto J. Math.).Google Scholar
  5. 5.
    Bogachev, V.I., Kolesnikov, A.V.: The Monge–Kantorovich problem: achievements, connections, and perspectives. Russ. Math. Surv. 67(5), 785–890 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Borell, C.: Convex measures on locally convex spaces. Ark. Math. 12(2), 239–252 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bourgain, J., Brezis, H., Mironescu, P.: Limiting embedding theorem for \(W^{s, p}\) when \(s \uparrow 1\) and applications. J. Anal. Math. 87, 77–101 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Caffarelli, L.: Interior \(W^{2, p}\) estimates for solutions of the Monge–Ampère equation. Ann. Math. 131(1), 35–150 (1990)CrossRefGoogle Scholar
  9. 9.
    Caffarelli, L., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society, Providence (1995)zbMATHGoogle Scholar
  10. 10.
    DePhilippis, G., Figalli, A.: \(W^{1,2}\) regularity for solutions of the Monge–Ampère equation. arXiv:1111.7207. 0000Google Scholar
  11. 11.
    DePhilippis, G., Figalli, A., Savin, O.: A note on the interior \(W^{2,1+\varepsilon }\) estimates for the Monge–Ampère equation. arXiv:1202.5566. 0000Google Scholar
  12. 12.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equation of the Second Order. Springer, Berlin (2001)Google Scholar
  13. 13.
    Gutiérrez, C.E.: The Monge–Ampère equation, Progress in Nonlinear Differential Equations and Their Applications, vol. 44. Birkhäuser, Brazil (2001)Google Scholar
  14. 14.
    Huang, Q.B.: On the mean oscillation of the Hessian of solutions to the Monge–Ampère equation. Adv. Math. 207, 599–616 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Kolesnikov, A.V.: On Sobolev regularity of mass transport and transportation inequalities, Theory of Probability and its Applications, to appear. Translated from Teor. Veroyatnost. i Primenen. 57(2), 296–321 (2012). Available online at arXiv:1007.1103Google Scholar
  16. 16.
    Kolesnikov, A.V.: Hessian structures and optimal transportation of log-concave measures. arXiv:1201. 2342. 0000Google Scholar
  17. 17.
    Kolyada, V.I., Lerner, A.K.: On limiting embeddings of Besov spaces. Stud. Math. 171(1), 1–13 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Krylov, N.V.: Fully nonlinear second order elliptic equations: recent developments. Ann. Scuola Norm. Sup. Pisa Cl. Sci. XXV(4), 569–595 (1997)Google Scholar
  19. 19.
    McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128(1), 153–179 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Mazya, V., Shaposhnikova, T.: On the Bourgain, Brezis and Mironescu theorem concerning limitiong embeddings of fractional Sobolev spaces. J. Func. Anal. 195, 230–238 (2002)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Pogorelov, A.V.: Monge–Ampère equations of elliptic type, Noordhoff (1964)Google Scholar
  22. 22.
    Savin, O.: Global \(W^{2, p}\) estimates for the Monge–Ampère equation. arXiv: 1103.0456. 0000Google Scholar
  23. 23.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University, Princeton (1973)Google Scholar
  24. 24.
    Schmidt, T.: \(W^{2,1+\varepsilon }\) estimates for the Monge–Ampère equation. http://cvgmt.sns.it/paper/1779
  25. 25.
    Trudinger, N.S., Wang, X.-J.: The Monge–Ampère equation and its geometric applications, Handbook of geometric analysis. Adv. Lect. Math. (ALM). 7(1), 467–524 (2008)Google Scholar
  26. 26.
    Villani, C.: Topics in Optimal Transportation. American Mathematical Society, Providence (2003)zbMATHGoogle Scholar
  27. 27.
    Villani, C.: Optimal Transport, Old and New, Vol. 338 of Grundlehren der mathematischen Wissenschaften. Springer, Berlin (2009)Google Scholar
  28. 28.
    Wang, X.-J.: Some counterexamples to the regularity of Monge–Ampère equations. Proc. Amer. Math. Soc. 123(3), 841–845 (1995)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alexander V. Kolesnikov
    • 1
  • Sergey Yu. Tikhonov
    • 2
    Email author
  1. 1.Higher School of EconomicsMoscowRussia
  2. 2.ICREA and Centre de Recerca MatemàticaBellaterraSpain

Personalised recommendations