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

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....

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Bibliographic Details
Published in:Journal of mathematical sciences (New York, N.Y.) N.Y.), 2020-07, Vol.248 (3), p.303-337
Main Author: Kokarev, V. N.
Format: Article
Language:English
Subjects:
Elliptic functions
Mathematical analysis
Mathematics
Mathematics and Statistics
Nonlinear equations
Polynomials
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Summary: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.
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-020-04874-2