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.
Similar content being viewed by others
References
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).
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).
R. Bellman, Introduction to Matrix Analysis, McGraw-Hill, New York etc. (1970).
W. Blaschke, Vorlesungen über Differentialgeometrie. II. Affine Differentialgeometrie, Springer-Verlag, Berlin (1923).
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).
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).
E. Calabi, “An extension of E. Hopf’s maximum principle with an application to Riemannian geometry,” Duke Math. J., 25, 45–56 (1958).
S. Y. Cheng and S. T. Yau, “Complete affine hypersurfaces. I. The completeness of affine metrics,” Commun. Pure Appl. Math., 39, 839–866 (1986).
A. Gray, Tubes, Addison-Wesley, Redwood City, CA (1990).
K. Jörgens, “¨Uber die Lösungen der Differentialgleichung rt−s2 = 1,” Math. Ann., 127, 130–134 (1954).
V. N. Kokarev, “Complete convex solutions of the equation spurm(zij) = 1,” Mat. Fiz. Anal. Geom., 3, No. 1/2, 102–117 (1996).
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).
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).
P. Lancaster, Theory of Matrices, Academic Press, New York–London (1969).
A. V. Pogorelov, Multidimensional Minkowski Problem [in Russian], Nauka, Moscow (1975).
A. V. Pogorelov, Multidimensional Monge–Ampère equation det(zij) = φ(z1,. . . , zn, z, x1, . . . , xn) [in Russian], Nauka, Moscow (1988).
G. Ţiţeica, “Sur one nouvelle classe de surfaces,” C. R. Acad. Sci. Paris, 145, 132–133 (1907).
G. Ţiţeica, “Sur one nouvelle classe de surfaces,” C. R. Acad. Sci. Paris, 146, 165–166 (1908).
V. L. Zaguskin, “Circumscribed and inscribes ellipsoids of extremal volume,” Usp. Mat. Nauk, 13, No. 6, 89–92 (1958).
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-04874-2