Построение решений уравнения типа Монжа - Ампера на основе ф-триангуляции

Construction of the solutions of the Monge - Ampere type equation based on f-triangulation

In the article we considered the method of geometric construction of piecewise linear analog solutions discrete form of the equation 𝑢𝑥1𝑥1𝑢𝑥2𝑥2 - 𝑢2 𝑥1𝑥2 = 𝐹(𝑢𝑥1, 𝑢𝑥2)'(𝑥1, 𝑥2). The idea of the method is based on the approach suggested by A.D. Aleksandrov to prove the existence of a classical solution of the above equation. Note that thegeometric analog of the problem being solved in this article is the problem of A.D. Aleksandrov on the existence of a polyhedron with prescribed curvatures of vertices. For piecewise linear convex function we defined curvature mesuare (𝑝𝑖) of vertex in terms of function 𝐹( 1, 2). The solution is defined as piecewise linear convex function with prescribed values (𝑝𝑖) = '𝑖, = 1,...,𝑁. The relation Φ-triangulations of given set of points 𝑖, = 1,...,𝑀 with piecewise linear solutions is obtained. The construction of solution is based on analog of Legendre transformation of kind 𝑓(𝑥) = min 𝑖=1,𝑀{Ψ( 𝑖) + ⟨∇Ψ( 𝑖), - 𝑖⟩}. As a corollary we proved the following result. Theorem 2. Let - classical Delaunay triangulation of a set of points -1,..., -𝑀 ∈ R2 with triangles Δ1,...,Δ𝑁 such that 𝐹(Δ𝑖) = '𝑖, = 1,...,𝑁. Then there is a piecewise linear function satisfying the equations (𝑝𝑖) = '𝑖, = 1,...,𝑁. Morever, the required solution 𝑓(𝑥) defined by 𝑓(𝑥) = min 𝑖=1,𝑀 {︂1 4|-𝑖|2 + ⟨-𝑖, - 1 2 -𝑖⟩}︂.