Skip to main content
SpringerLink
Log in

Regularity of Degenerate Monge–Ampère and Prescribed Gaussian Curvature Equations in Two Dimensions

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

We use a priori inequalities for quasilinear equations to obtain a regularity theorem for the Dirichlet problem for the Monge–Ampère equation,

$$u_{xx}u_{yy}-(u_{xy})^{2}=k(x,y),$$

and the prescribed Gaussian curvature equation,

$$u_{xx}u_{yy}-(u_{xy})^{2}=k(x,y)(1+u_{x}^{2}+u_{y}^{2})^{2},$$

where k(x,y) is close to a function of one variable alone when k is small, but permitted to vanish to infinite order.

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

Access this article

Log in via an institution

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. Alexandrov, A.D.: ‘Dirichlet's problem for the equation Det ‖z ij‖=Φ(z 1,. . .,z n,z,x 1,. . .,x n), I’, Vestnik Leningrad Univ. Ser. Mat. Mekh. Astr. 13 (1958), 5–24.

    Google Scholar 

  2. Bedford, E. and Fornaess, J.E.: ‘Counterexamples to regularity for the complex Monge–Ampère equation’, Invent. Math. 50 (1979), 129–134.

    Article  MathSciNet  MATH  Google Scholar 

  3. Caffarelli, L. and Cabré, L.: Fully Nonlinear Elliptic Equations, Colloq. Publ. 43, Amer. Math. Soc., Providence, 1995.

  4. Caffarelli, L., Nirenberg, L. and Spruck, J.: ‘The Dirichlet problem for nonlinear second order elliptic equations, I. Monge–Ampère equations’, Comm. Pure Appl. Math. 37 (1984), 369–402.

    MathSciNet  MATH  Google Scholar 

  5. Cheng, S.-Y. and 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 (1977), 41–68.

    MathSciNet  MATH  Google Scholar 

  6. Christ, M.: ‘Hypoellipticity in the infinitely degenerate regime’, Preprint on website.

  7. Fedi \(\mbox{\u{\i}}\) , V.S.: ‘On a criterion for hypoellipticity’, Math. USSR-Sb. 14 (1971), 15–45.

  8. Franchi, B.: ‘Weighted Sobolev–Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations’, Trans. Amer. Math. Soc. 327 (1991), 125–158.

    Article  MATH  MathSciNet  Google Scholar 

  9. Gilbarg, D. and Trudinger, N.: Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, revised 3rd printing, 1998.

  10. Guan, P.: ‘Regularity of a class of quasilinear degenerate elliptic equations’, Adv. Math. 132 (1997), 24–45.

    Article  MATH  MathSciNet  Google Scholar 

  11. Guan, P.: ‘C 2 a priori estimates for degenerate Monge–Ampère equations’, Duke Math. J. 86 (1997), 323–346.

    Article  MATH  MathSciNet  Google Scholar 

  12. Guan, P., Trudinger, N.S. and Wang, X.-J.: ‘On the Dirichlet problem for degenerate Monge–Ampère equations’, Acta Math. 182 (1999), 87–104.

    MathSciNet  MATH  Google Scholar 

  13. Heinz, E.: ‘On elliptic Monge–Ampère equations and Weyl's embedding problem’, J. Anal. Math. 7 (1959), 1–52.

    Article  MATH  MathSciNet  Google Scholar 

  14. Heinz, E.: ‘Interior estimates for solutions of elliptic Monge–Ampère equations’, Proc. Sympos. Pure Math. 4 (1960), 149–155.

    Google Scholar 

  15. Ivochkina, N.: ‘Solution of the Dirichlet problem for curvature equations of order m’, Math. USSR-Sb. 67 (1990), 317–339.

    Article  MATH  MathSciNet  Google Scholar 

  16. Kohn, J.J.: ‘Hypoellipticity of some degenerate subelliptic operators’, Preprint, 1997, 1–13.

  17. Ladyzhenskaya, O.A. and Ural'tseva, N.N.: Linear and Quasilinear Elliptic Equations, Nauka, Moscow, 1965 (Russian). English transl. Academic Press, New York, 1968. 2nd Russian edn, 1973.

    Google Scholar 

  18. Lewy, H.: ‘A priori limitations for solutions of Monge–Ampère equations I, II’, Trans. Amer. Math. Soc. 37 (1935), 417–434.

    Article  MATH  MathSciNet  Google Scholar 

  19. Lewy, H.: ‘On the non-vanishing of the Jacobian in certain one-to-one mappings’, Bull. Amer. Math. Soc. 42 (1936), 689–692.

    Article  MATH  MathSciNet  Google Scholar 

  20. Moser, J.: ‘On Harnack's theorem for elliptic differential equations’, Comm. Pure Appl. Math. 14 (1961), 577–591.

    MATH  MathSciNet  Google Scholar 

  21. Nirenberg, L.: ‘The Weyl and Minkowski problems in differential geometry in the large’, Comm. Pure Appl. Math. 6 (1953), 337–394.

    MATH  MathSciNet  Google Scholar 

  22. Pogorelov, A.V.: ‘The regularity of a convex surface with a given Gauss curvature’, Mat. Sb. (N.S.) 31 (1952), 88–103 (Russian).

    MATH  MathSciNet  Google Scholar 

  23. Rios, C., Sawyer, E. and Wheeden, R.L.: ‘A higher dimensional partial Legendre transform, and regularity of degenerate Monge–Ampère equations’, Adv. in Math. 193 (2005), 373–415.

    Article  MathSciNet  MATH  Google Scholar 

  24. Rios, C., Sawyer, E. and Wheeden, R.L.: ‘A priori estimates for infinitely degenerate quasilinear equations’, Preprint.

  25. Rios, C., Sawyer E. and Wheeden, R.L.: ‘Hypoellipticity for infinitely degenerate quasilinear equations and the Dirichlet problem’, Preprint.

  26. Sawyer, E.: ‘A symbolic calculus for rough pseudodifferential operators’, Indiana Univ. Math. J. 45 (1996), 289–332.

    Article  MATH  MathSciNet  Google Scholar 

  27. Sawyer, E. and Wheeden, R.L.: ‘A priori estimates for quasilinear equations related to the Monge–Ampère equation in two dimensions’, J. Anal. Math., to appear.

  28. Sawyer, E. and Wheeden, R.L.: ‘Regularity of degenerate Monge–Ampère and prescribed Gaussian curvature equations in two dimensions’, Preprint available at http://www.math.mcmaster.ca/~sawyer.

  29. Schulz, F.: Regularity Theory for Quasilinear Elliptic Systems and Monge–Ampère Equations in Two Dimensions, Lecture Notes in Math. 1445, Springer-Verlag, New York, 1990.

  30. Trudinger, N. and Urbas, J.: ‘The Dirichlet problem for the equation of prescribed Gauss curvature’, Bull. Austral. Math. Soc. 28 (1983), 217–231.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. McMaster University, Hamilton, Ontario

    Eric T. Sawyer

  2. Rutgers University, New Brunswick, NJ

    Richard L. Wheeden

Authors
  1. Eric T. Sawyer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Richard L. Wheeden
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric T. Sawyer.

Additional information

Mathematics Subject Classifications (2000)

35B65, 35B45.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sawyer, E.T., Wheeden, R.L. Regularity of Degenerate Monge–Ampère and Prescribed Gaussian Curvature Equations in Two Dimensions. Potential Anal 24, 267–301 (2006). https://doi.org/10.1007/s11118-005-0915-4

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-005-0915-4

Keywords

Search

Navigation