Advertisement

W2, p Estimates for the Monge–Ampère Equation

  • Cristian E. Gutiérrez
Chapter
  • 1k Downloads
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 89)

Abstract

Our purpose in this chapter is to prove Caffarelli’s interior L p estimates for second derivatives of solutions to the Monge–Ampère equation. That is, solutions u to Mu = f with f positive and continuous have second derivatives in L p , for 0 < p < , Theorem 6.4.2.

Bibliography

  1. [Caf90a]
    L.A. Caffarelli, Interior W 2, p estimates for solutions of the Monge–Ampère equation. Ann. Math. 131, 135–150 (1990)Google Scholar
  2. [CG96]
    L.A. Caffarelli, C.E. Gutiérrez, Real analysis related to the Monge–Ampère equation. Trans. Am. Math. Soc. 348 (3), 1075–1092 (1996)CrossRefzbMATHGoogle Scholar
  3. [CH62]
    R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1962)Google Scholar
  4. [CY77]
    S.-Y. Cheng, S.-T. Yau, On regularity of the Monge–Ampère equation \(\det ( \dfrac{\partial ^{2}u} {\partial x_{i}\partial x_{j}}) = f(x,u)\). Commun. Pure Appl. Math. XXX, 41–68 (1977)Google Scholar
  5. [FGL08]
    G. Di Fazio, C.E. Gutiérrez, E. Lanconelli, Covering theorems, inequalities on metric spaces and applications to PDE’s. Math. Ann. 341 (2), 255–291 (2008)Google Scholar
  6. [GH01]
    C.E. Gutiérrez, Q. Huang, W 2, p-estimates for the parabolic Monge–Ampère equation. Arch. Ration. Mech. Anal. 159 (2), 137–177 (2001)Google Scholar
  7. [Gut]
    C.E. Gutiérrez, A covering lemma of Besicovitch type on the Heisenberg group \(\mathbb{H}^{n}\) (1998). unpublished manuscriptGoogle Scholar
  8. [Hua09]
    Q. Huang, Sharp regularity results on second derivatives of solutions to the Monge-Ampère equation with VMO type data. Commun. Pure App. Math. 62 (5), 677–705 (2009)CrossRefzbMATHGoogle Scholar
  9. [PFS13]
    G. De Philippis, A. Figalli, O. Savin, A note on interior W 2, 1+ε estimates for the Monge-Ampère equation. Math. Ann. 357 (1), 11–22 (2013)Google Scholar
  10. [Pog71]
    A.V. Pogorelov, On the regularity of generalized solutions of the equation \(\det ( \dfrac{\partial ^{2}u} {\partial x^{i}\partial x^{j}}) =\phi (x^{1},\ldots,x^{n})> 0\). Sov. Math. Dokl. 12 (5), 1436–1440 (1971)Google Scholar
  11. [Sav13a]
    O. Savin, Global W 2, p estimates for the Monge-Ampère equation. Proc. Am. Math. Soc. 141 (10), 3573–3578 (2013)Google Scholar
  12. [Sch93]
    R. Schneider, Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 44 (Cambridge University Press, Cambridge, 1993)Google Scholar
  13. [Ste93]
    E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43 (Princeton University Press, Princeton, 1993)Google Scholar
  14. [Urb88]
    J.I.E. Urbas, Regularity of generalized solutions of Monge–Ampère equations. Math. Z. 197, 365–393 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [Wan92]
    X.-J. Wang, Remarks on the regularity of Monge–Ampère equations, in Proceedings of the International Conference on Nonlinear PDE, Hangzhou (Academic, Beijing, 1992)Google Scholar
  16. [Wan95]
    X.-J. Wang, Some counterexamples to the regularity of Monge–Ampère equations. Proc. Am. Math. Soc. 123 (3), 841–845 (1995)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Cristian E. Gutiérrez
    • 1
  1. 1.Department of MathematicsTemple UniversityPhiladelphiaUSA

Personalised recommendations