Mathematische Zeitschrift

, Volume 197, Issue 3, pp 365–393 | Cite as

Regularity of generalized solutions of Monge-Ampère equations

  • John I. E. Urbas
Article

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aleksandrov, A.D.: Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it. Učen. Zap. Leningrad. Gos. Univ.37, Mat. 3, 3–35, (1939) [Russian]Google Scholar
  2. 2.
    Aleksandrov, A.D.: Smoothness of a convex surface of bounded Gaussian curvature. Dokl. Akad. Nauk. SSSR36, 195–199 (1942) [Russian]Google Scholar
  3. 3.
    Aleksandrov, A.D.: Die innere Geometrie der konvexen Flächen. Berlin: Akademie-Verlag, 1955Google Scholar
  4. 4.
    Aleksandrov, A.D.: Dirichlet's problem for the equation Det |z ij|=Ψ(z 1, ...,z n,z, x 1, ...,x n). Vestn. Leningr. Univ.18, 5–29 (1963) [Russian]Google Scholar
  5. 5.
    Aleksandrov, A.D.: Uniqueness conditions and estimates for the solution of the Dirichlet problem. Vestn. Leningr. Univ.18, 5–29 (1963) [Russian]. English translation in Am. Math. Soc. Transl. (2)68, 89–119 (1968)Google Scholar
  6. 6.
    Aleksandrov, A.D.: Majorization of solutions of second-order linear equations, Vestn. Leningr. Univ.21, 5–25, (1966) [Russian]. English translation in Am. Math. Soc. Transl. (2)68, 120–143 (1968)Google Scholar
  7. 7.
    Bakel'man, I.Ya.: The Dirichlet problem for the ellipticn-dimensional Monge-Ampère equations and related problems in the theory of quasilinear equations, Proceedings of Seminar on Monge-Ampère Equations and Related Topics (Firenze 1980), Instituto Nazionale di Alta Matematica, Roma, 1–78 (1982)Google Scholar
  8. 8.
    Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations. I. Monge-Ampère equation. Commun. Pure Appl. Math.37, 369–402 (1984)Google Scholar
  9. 9.
    Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for the degenerate Monge-Ampère equation. (To appear)Google Scholar
  10. 10.
    Cheng, S.-Y., Yau, S.T.: On the regularity of the Monge-Ampère equation det (∂2 u/∂x ix j) =F(x, u). Commun. Pure Appl. Math.30, 41–68 (1977)Google Scholar
  11. 11.
    Donaldson, T.K., Trudinger, N.S.: Orlicz-Sobolev spaces and imbedding theorems. J. Funct. Anal.8, 52–75 (1971)Google Scholar
  12. 12.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, Second Edition. Berlin-Heidelberg-New York-Tokyo: Springer 1983Google Scholar
  13. 13.
    Heinz, E.: Über die Differentialungleichung 0<α≦rts 2≦β<∞. Math. Z.72, 107–126 (1959)Google Scholar
  14. 14.
    Ivochkina, N.M.: Construction of a priori bounds for convex solutions of the Monge-Ampère equation by integral methods. Ukrain. Mat. Ž.30, 45–53 (1978) [Russian]Google Scholar
  15. 15.
    Ivochkina, N.M.: Classical solvability of the Dirichlet problem for the Monge-Ampère equation. Zap. Naučn. Semin. Leningr. Otd. Mat. Inst. Steklova (LOMI)131, 72–79 (1983) [Russian]Google Scholar
  16. 16.
    Jensen, R.: The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. (To appear)Google Scholar
  17. 17.
    Kufner, A., John, O., Fučik, S.: Function spaces, Leyden: Noordhoff 1977Google Scholar
  18. 18.
    Nikolaev, I.G., Shefel', S.Z.: Convex surfaces with positive bounded specific curvature and a priori estimates for Monge-Ampère equations. Sib. Mat. Ž.26, 120–136 (1985) [Russian]Google Scholar
  19. 19.
    Pogorelov, A.V.: Monge-Ampère equations of elliptic type, Groningen: Noordhoff 1964Google Scholar
  20. 20.
    Pogorelov, A.V.: On the regularity of generalized solutions of the equation det(∂2 u/∂x ix j) = ψ(x 1, ...,x n) > 0. Dokl. Akad. Nauk. SSSR200, 543–547 (1971) [Russian]. English translation in Soviet Math. Dokl.12, 1436–1440 (1971)Google Scholar
  21. 21.
    Pogorelov, A.V.: The Dirichlet problem for then-dimensional analogue of the Monge-Ampère equation. Dokl. Akad. Nauk. SSSR201, 790–793 (1971) [Russian]. English translation in Soviet Math. Dokl.12, 1727–1731 (1971)Google Scholar
  22. 22.
    Pogorelov, A.V.: Extrinsic geometry of convex surfaces. Providence: American Mathematical Society 1973Google Scholar
  23. 23.
    Pogorelov, A.V.: The Minkowski multidimensional problem. New York: J. Wiley 1978Google Scholar
  24. 24.
    Pogorelov, A.V.: Regularity of generalized solutions of the equation det (u ij)π(▽u, u, x)=Ψ(x). Dokl. Akad. Nauk. SSSR275, 26–28 (1984), [Russian]. English translation in Soviet Math. Dokl.29, 159–161 (1984)Google Scholar
  25. 25.
    Rauch, J., Taylor, B.A.: The Dirichlet problem for the multi-dimensional Monge-Ampère equation, Rocky Mt. J. Math.7, 345–364 (1977)Google Scholar
  26. 26.
    Sabitov, I.Kh.: The regularity of convex regions with a metric that is regular in the Hölder classes. Sibirsk. Mat. Ž.17, 907–915 (1976) [Russian]. English translation in Sib. Math. J.17, 681–687 (1976)Google Scholar
  27. 27.
    Schulz, F.: Über die Differentialgleichungrts 2=f und das Weylsche Einbettungsproblem. Math. Z.179, 1–10 (1982)Google Scholar
  28. 28.
    Schulz, F.: Über nichtlineare, konkave elliptische Differentialgleichungen. Math. Z.191, 429–448 (1986)Google Scholar
  29. 29.
    Trudinger, N.S.: Regularity of solutions of fully nonlinear elliptic equations. Boll. Unione Mat. Ital. (6)3-A, 421–430 (1984)Google Scholar
  30. 30.
    Trudinger, N.S., Urbas, J.I.E.: On second derivative estimates for equations of Monge-Ampère type. Bull. Aust. Math. Soc.30, 321–334 (1984)Google Scholar
  31. 31.
    Urbas, J.I.E.: The equation of prescribed Gauss curvature without boundary conditions. J. Differ. Geom.20, 311–327 (1984)Google Scholar
  32. 32.
    Urbas, J.I.E.: The generalized Dirichlet problem for equations of Monge-Ampère type. Ann. Inst. Henri Poincaré, Analyse Non Linéaire3, 209–228 (1986)Google Scholar
  33. 33.
    Wang, R.H., Jiang, J.: Another approach to the Dirichlet problem for equations of Monge-Ampère type. Northeastern Math. J.1, 27–40 (1985)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • John I. E. Urbas
    • 1
  1. 1.Centre for Mathematical AnalysisAustralian National UniversityCanberraAustralia

Personalised recommendations