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A priori estimates and a Liouville theorem for complex Monge-Ampère equations

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Bibliography

  1. Aubin, T.: Nonlinear analysis on manifolds. Monge-Ampère equations. Berlin-Heidelberg-New York: Springer 1982

    Google Scholar 

  2. Bedford, E., Taylor, B.A.: The Dirichlet problem for an equation of complex Monge-Ampère type. In: Partial Differential Equations and Geometry. Proceedings of the Park City Conference (Park City, Utah 1977), edited by C.I. Byrnes, pp. 39–50. Basel-New York: Dekker 1979

    Google Scholar 

  3. Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations. To appear

  4. Calabi, E.: Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens. Michigan Math. J.5, 105–126 (1958)

    Google Scholar 

  5. Cheng, S.-Y., Yau, S.T.: On the regularity of the Monge-Ampère equation det (∂2 u/∂x i x j ) Comm. Pure Appl. Math.30, 41–68 (1977)

    Google Scholar 

  6. Cheng, S.-Y., Yau, S.-T.: On the existence of a complexe Kähler metric on non-compact complex manifolds and the regularity of Fefferman's equations. Comm. Pure Appl. Math.33, 507–544 (1980)

    Google Scholar 

  7. Fefferman, C.: Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains. Ann. of Math. (2)103, 395–416 (1976)

    Google Scholar 

  8. Gaveau, B.: Solutions of Monge-Ampère equations by optimal control methods and applications. In: Partial Differential Equations and Geometry. Proceedings of the Park City Conference (Park City, Utah 1977), edited by C.I. Byrnes, pp. 10–17. Basel-New York: Dekker 1979

    Google Scholar 

  9. Gilbarg, D., Trudinger, N.S.: Elliptic partials differential equations of second order. Berlin-Heidelberg-New York: Springer 1977

    Google Scholar 

  10. Ivochkina, N.M.: Construction of a priori bounds for convex solutions of the Monge-Ampère equation by integral methods. Ukrainian Math. J.30, 32–38 (1978)

    Google Scholar 

  11. Lions, P.-L.: Sur les equations de Monge-Ampère, I. Manuscripta Math.41, 1–43 (1983)

    Google Scholar 

  12. Meyers, N.G.: On a class of nonuniformly elliptic quasi-linear equations in the plane. Arch. Rational Mech. Anal.12, 367–391 (1963)

    Google Scholar 

  13. Oskolkov, A.P.: Some estimates for nonuniformly elliptic equations and systems. Proc. Steklov Inst. Math.92, 233–267 (1966)

    Google Scholar 

  14. Pogorelov, A.V.: The Minkowski multidimensional problem. Washington, D.C.: Winston 1978

    Google Scholar 

  15. Schulz, F.: A priori estimates and a Liouville theorem for elliptic Monge-Ampère equations. Math. Ann. (to appear)

  16. Schulz, F.: AC 2-estimate for solutions of complex Monge-Ampère equations. J. Reine Angew. Math. (to appear)

  17. Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I. Comm. Pure Appl. Math.31, 339–411 (1978)

    Google Scholar 

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The second author was partially sypported by the Sonderforschungsbereich 72 at the University of Bonn

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Riebesehl, D., Schulz, F. A priori estimates and a Liouville theorem for complex Monge-Ampère equations. Math Z 186, 57–66 (1984). https://doi.org/10.1007/BF01215491

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  • DOI: https://doi.org/10.1007/BF01215491

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