Skip to main content

W 2,1 regularity for solutions of the Monge–Ampère equation

Abstract

In this paper we prove that a strictly convex Alexandrov solution u of the Monge–Ampère equation, with right-hand side bounded away from zero and infinity, is \(W^{2,1}_{\mathrm{loc}}\). This is obtained by showing higher integrability a priori estimates for D 2 u, namely D 2 uLlogk L for any k∈ℕ.

This is a preview of subscription content, access via your institution.

References

  1. Ambrosio, L.: Transport Equation and Cauchy Problem for Non-Smooth Vector Fields. In: Calculus of Variations and Nonlinear Partial Differential Equations. Lecture Notes in Math., vol. 1927, pp. 1–41. Springer, Berlin (2008)

    Chapter  Google Scholar 

  2. Ambrosio, L., De Philippis, G., Kirchheim, B.: Regularity of optimal transport maps and partial differential inclusions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 22(3), 311–336 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  3. Brenier, Y.: Décomposition polaire et réarrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris Sér. I Math. 305(19), 805–808 (1987)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  5. Caffarelli, L.: A localization property of viscosity solutions to the Monge–Ampère equation and their strict convexity. Ann. Math. (2) 131(1), 129–134 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  6. Caffarelli, L.: Interior W 2,p estimates for solutions of the Monge–Ampère equation. Ann. Math. (2) 131(1), 135–150 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  7. Caffarelli, L.: Some regularity properties of solutions to Monge–Ampére equations. Commun. Pure Appl. Math. 44, 965–969 (1991)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  9. Caffarelli, L., Gutierrez, C.: Real analysis related to the Monge–Ampère equation. Trans. Am. Math. Soc. 348, 1075–1092 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  10. Figalli, A., Kim, Y.-H.: Partial regularity of Brenier solutions of the Monge–Ampr̀e equation. Discrete Contin. Dyn. Syst., Ser. A 28(2), 559–565 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  11. Gutierrez, C.: The Monge–Ampére Equation. Progress in Nonlinear Differential Equations and Their Applications, vol. 44. Birkhäuser, Boston (2001)

    Book  MATH  Google Scholar 

  12. Gutierrez, C., Huang, Q.: Geometric properties of the sections of solutions to the Monge–Ampère equation. Trans. Am. Math. Soc. 352(9), 4381–4396 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  13. John, F.: Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, pp. 187–204. Interscience, New York (1948)

    Google Scholar 

  14. Savin, O.: A localization property at the boundary for the Monge–Ampère equation. Preprint (2010)

  15. Savin, O.: Global W 2,p estimates for the Monge–Ampère equations. Preprint (2010)

  16. Stein, E.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. In: With the Assistance of Timothy S. Murphy. Monographs in Harmonic Analysis, III. Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton (1993)

    Google Scholar 

  17. Urbas, J.I.E.: Regularity of generalized solutions of Monge–Ampère equations. Math. Z. 197(3), 365–393 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  18. Wang, X.-J.: Some counterexamples to the regularity of Monge–Ampère equations. Proc. Am. Math. Soc. 123(3), 841–845 (1995) (English summary)

    MATH  Google Scholar 

  19. Zygmund, A.: Trigonometric Series, 2nd edn., vol. I. Cambridge University Press, New York (1959)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alessio Figalli.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

De Philippis, G., Figalli, A. W 2,1 regularity for solutions of the Monge–Ampère equation. Invent. math. 192, 55–69 (2013). https://doi.org/10.1007/s00222-012-0405-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-012-0405-4

Keywords

  • Differential Inclusion
  • High Integrability
  • Doubling Measure
  • Minkowski Problem
  • Bounded Convex Domain