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BSU bulletin. Mathematics, Informatics

Bibliographic description:
Filimonova A. P.
,
Yuryeva T. A.
PRIOR ESTIMATES OF THE GRADIENT FOR SOLUTION OF A CERTAIN MONGE—AMPÈRE EQUATION // BSU bulletin. Mathematics, Informatics. - 2019. №1. . - С. 49-55.
Title:
PRIOR ESTIMATES OF THE GRADIENT FOR SOLUTION OF A CERTAIN MONGE—AMPÈRE EQUATION
Financing:
Codes:
DOI: 10.18101/2304-5728-2019-1-49-55UDK: 517.953
Annotation:
The resolution of the issue of existence and uniqueness of surfaces with given geo- metric characteristics in various spaces is associated with finding prior estimates for solution of a nonlinear Monge—Ampère differential equation in the corresponding metrics. Such geometric characteristics include Gaussian curvature, average curva- ture, the sum of principal radii of curvature, etc. The article describes surfaces ho- meomorphic to the sphere of a unit radius from the class of regular convex in three- dimensional space of constant negative curvature with a given function of intrinsic (Gaussian) curvature. Intrinsic curvature is considered as a function of the point of three-dimensional Lobachevsky space. The solution for a Monge—Ampère differ- ential equation is assumed to be a function explicitly given in spherical coordinates. The article describes the procedure for constructing prior estimates of the first de- rivatives of equation solution. It is assumed the availability of estimates of the solu- tion itself.
Keywords:
hyperbolic space; Monge—Ampère equation; negative ellipticity; Beltrami coordinates; Gaussian curvature.
List of references:
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Filimonova A. P., Yuryeva T. A. Apriornye otsenki resheniya v metrike urav- neniya tipa Monzha—Ampera na sfere kak dvumernom mnogoobrazii v prostranstve postoyannoi krivizny [Prior Estimates of the Solution of a Monge—Ampére Type Equation in С 0 S 2  Metrics on a Sphere as Two-Dimensional Manifold in Constant Curvature Space]. Mezhdunarodnyi nauchno-issledovatelskii zhurnal. 2016. No. 9– 2(51). Pp. 132–136.

Filimonova A. P., Yuryeva T. A. Svoistvo vypuklosti funktsii vneshnei krivizny poverkhnosti v trekhmernom prostranstve Lobachevskogo [A Convexity Property of the Function of Surface External Curvature in Three-Dimensional Lobachevsky Space]. Vestnik Amurskogo gosudarstvennogo universiteta. 2015. V. 69. Pp. 22–25.

Pogorelov A. V. Mnogomernoe uravnenie Monzha—Ampera [Multidimensional Monge—Ampère Equation]. Moscow: Nauka Publ., 1988. 96 p.