Skip to main content
SpringerLink
Log in

The regularity of the surface of the Gauss curvature 0≤K ∈ C ∞0

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

Under the mild conditions, it is proved that the convex surface is global C1.1, with the given Gaussian curvature 0≤K ∈ C ∞0 and the given boundary curve. Examples are given to show that the regularity is optimal.

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. Pogorelov, A. V., The regularity of a convex surface with a given Gauss curvature,Mat. Sb., 1952, 31: 88.

    MathSciNet  Google Scholar 

  2. Nirenberg, L., The Weyl and Minkowski problems in differential geometry in the large,Comm. Pure Appl. Math., 1953, 6: 337.

    Article  MATH  MathSciNet  Google Scholar 

  3. Cheng, S. Y., Yau, S. T., On the regularity of the solution of the n -dimensional Minkowski problem.Comm. Pure Appl. Math., 1976, 29: 495.

    Article  MATH  MathSciNet  Google Scholar 

  4. Trudinger, N. S., Urbas. J. I. E., The Dirichlet problem for the equation of prescribed Gauss curvature,Bull. Austral. Math. Soc., 1983, 30: 217.

    Article  MathSciNet  Google Scholar 

  5. Bedford, E., Fornaess. J., Counterexamples to regularity for the complex Monge-Ampére equation,Inv. Math., 1979, 50: 129.

    Article  MATH  MathSciNet  Google Scholar 

  6. Caffarelli, L. A., Nirenberg, L., Spruck. J., The Dirichlet problem for degenerate Monge-Ampére equation,Revista Mathematica Iberoamericana, 1986, 2: 19.

    MathSciNet  Google Scholar 

  7. Hong, J. X., Dirichlet problems for Monge- Ampére equation degenerate on boundary,Chin. Ann. of Math., 1991, l2B: 407.

    Google Scholar 

  8. Hong, J. X., The estimates near the boundary for solutions of Monge- Ampére equations.J. Partial Diff. Eqs., 1994, 7: 97.

    MATH  Google Scholar 

  9. Kutev, N., On the solvability of the Dirichlet problem for degenerate equations of Monge- Ampére type,Nonlinear Analysis, Theory and Applications, 1989, 13: 1475.

    Article  MATH  MathSciNet  Google Scholar 

  10. Caffarelli, L. A., Cabre. X.,Fully Nonlineur Elliptic Equations, Providence: American Mathematical Society, 1995.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Pcking University, 100871, Beijing, China

    Jiguang Bao

  2. Department of Mathematics, Beijing Normal University, 100875, Beijing, China

    Jiguang Bao

Authors
  1. Jiguang Bao
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Project supported by the Doctoral Funds of China and the National Natural Science Foundation of China (Grant No. 19771009).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bao, J. The regularity of the surface of the Gauss curvature 0≤K ∈ C ∞0 . Sci. China Ser. A-Math. 41, 1047–1050 (1998). https://doi.org/10.1007/BF02871838

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02871838

Keywords

Search

Navigation