Solving the Monge-Ampere Equation using Geometric Transformations
Solving the Monge-Ampere Equation using Geometric Transformations
DOI:
https://doi.org/10.56143/ujmcs.v1i2.5/Keywords:
Galilean motion; Monge-Ampere equation; Heisenberg group; total curvature; polar coordinat system.Abstract
The geometric problem of recovering a convex surface from a given function is equivalent to solving a certain Monge-Ampère equation. In this case, the extrinsic curvature is defined as a function of Borel sets. I. Ya. Bakelman constructed this theory and proved the existence and uniqueness of the solution of the Monge-Ampère equation of elliptic type in a simply connected convex domain. A. Artykbaev generalized this solution for a non-simply connected domain applying of the geometry of Galilean space. This paper is devoted to the analytical solution of the Monge-Ampère equation in a non-simply connected domain. The extrinsic curvature of the surface is determined in a non-simply connected domain which is bounded by concentric circles. By applying the transformation which is the motion of Galilean space and the transition to the polar coordinate system, the equation is modified, in which it is possible to separate the variables of the solution, the equation is sought for the sum of three functions. As a result, an analytical form of the solution in a non-simply connected domain bounded by concentric circles is obtained.
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