Complete Convex Solutions of Monge–Ampère-Type Equations and their Analogs

Abstract

In this paper, we study complete convex solutions of certain nonlinear elliptic equations by using geometric methods. We present a proof of the Jörgens–Calabi–Pogorelov theorem about improper convex affine spheres based on the study of complete convex solutions of the simplest Monge–Ampère equation. We consider a similar problem for Monge–Ampère equations of more general types. We prove that, under certain assumptions, solutions of these equations are quadratic polynomials.

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References

  1. 1.

    S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I,” Commun. Pure Appl. Math., 12, 623–727 (1959).

    MathSciNet  Article  Google Scholar 

  2. 2.

    A. D. Aleksandrov, “On the theory of mixrd volumes of convex bodies. IV. Mixed discriminants and mixed volumes,” Mat. Sb., 3, No. 2, 227–251 (1938).

    MathSciNet  Google Scholar 

  3. 3.

    R. Bellman, Introduction to Matrix Analysis, McGraw-Hill, New York etc. (1970).

    Google Scholar 

  4. 4.

    W. Blaschke, Vorlesungen über Differentialgeometrie. II. Affine Differentialgeometrie, Springer-Verlag, Berlin (1923).

  5. 5.

    L. Caffarelli, L. Nirenberg, and J. Spruck, “The Dirichlet problem for nonlinear second order elliptic equations. III. Functions of the eigenvalues of Hessian,” Acta Math., 155, Nos. 3, 4, 261–304 (1985).

  6. 6.

    E. Calabi, “Improper affine hyperspheres of convex type and a generalizations of a theorem by K. Jörgens,” Michigan Math. J., 5, No. 2, 105–126 (1958).

    MathSciNet  Article  Google Scholar 

  7. 7.

    E. Calabi, “An extension of E. Hopf’s maximum principle with an application to Riemannian geometry,” Duke Math. J., 25, 45–56 (1958).

    MathSciNet  Article  Google Scholar 

  8. 8.

    S. Y. Cheng and S. T. Yau, “Complete affine hypersurfaces. I. The completeness of affine metrics,” Commun. Pure Appl. Math., 39, 839–866 (1986).

    MathSciNet  Article  Google Scholar 

  9. 9.

    A. Gray, Tubes, Addison-Wesley, Redwood City, CA (1990).

  10. 10.

    K. Jörgens, “¨Uber die Lösungen der Differentialgleichung rt−s2 = 1,” Math. Ann., 127, 130–134 (1954).

    MathSciNet  Article  Google Scholar 

  11. 11.

    V. N. Kokarev, “Complete convex solutions of the equation spurm(zij) = 1,” Mat. Fiz. Anal. Geom., 3, No. 1/2, 102–117 (1996).

    MathSciNet  Google Scholar 

  12. 12.

    V. N. Kokarev, “On the equation of an improper convex affine sphere: a generalization of a theorem of Jörgens,” Mat. Sb., 194, No. 11, 65–80 (2003).

    MathSciNet  Article  Google Scholar 

  13. 13.

    V. N. Kokarev, “On complete convex solutions of equations similar to the improper affine sphere equation,” J. Math. Phys. Anal. Geom., 3, No. 4, 448–467 (2007).

    MATH  Google Scholar 

  14. 14.

    P. Lancaster, Theory of Matrices, Academic Press, New York–London (1969).

    Google Scholar 

  15. 15.

    A. V. Pogorelov, Multidimensional Minkowski Problem [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  16. 16.

    A. V. Pogorelov, Multidimensional Monge–Ampère equation det(zij) = φ(z1,. . . , zn, z, x1, . . . , xn) [in Russian], Nauka, Moscow (1988).

  17. 17.

    G. Ţiţeica, “Sur one nouvelle classe de surfaces,” C. R. Acad. Sci. Paris, 145, 132–133 (1907).

    Google Scholar 

  18. 18.

    G. Ţiţeica, “Sur one nouvelle classe de surfaces,” C. R. Acad. Sci. Paris, 146, 165–166 (1908).

    Google Scholar 

  19. 19.

    V. L. Zaguskin, “Circumscribed and inscribes ellipsoids of extremal volume,” Usp. Mat. Nauk, 13, No. 6, 89–92 (1958).

    MathSciNet  Google Scholar 

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Correspondence to V. N. Kokarev.

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 147, Proceedings of the Workshop on Algebra and Geometry of the Samara University, 2018.

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Kokarev, V.N. Complete Convex Solutions of Monge–Ampère-Type Equations and their Analogs. J Math Sci 248, 303–337 (2020). https://doi.org/10.1007/s10958-020-04874-2

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Keywords and phrases

  • improper convex affine sphere
  • Monge-Ampère equation

AMS Subject Classification

  • 58J05