Construction of the solutions of the Monge–Ampere type equation based on $\Phi$-triangulation

In the article we considered the method of geometric construction of piecewise linear analog solutions discrete form of the equation
$$u_{x_1x_1} u_{x_2x_2} -u_ {x_1x_2}^ 2 = F (u_{x_1}, u_{x_2}) \varphi (x_1, x_2).$$
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 the geometric 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 $\mu(p_i)$ of vertex $p_i$ in terms of function $F(\xi_1,\xi_2)$. The solution is defined as piecewise linear convex function with prescribed values $\mu(p_i)=\varphi_i, i=1,...,N$. The relation $\Phi$-triangulations of given set of points $\xi_i,i=1,...,M$ with piecewise linear solutions is obtained. The construction of solution is based on analog of Legendre transformation of kind
$$f(x) = \min_{i = \overline{1,M}} \{ \Psi(\xi_i) + \left\langle \nabla \Psi (\xi_i) , x - \xi_i \right\rangle \}.$$
As a corollary we proved the following result.
Theorem 2. Let $T$—classical Delaunay triangulation of a set of points ${\eta} _ {1}, ..., {\eta} _ {M} \in \mathbb {R}^2$ with triangles $\Delta_1 ,. .., \Delta_N$ such that $\mu_F (\Delta_i) = \varphi_i, i=1,...,N$. Then there is a piecewise linear function satisfying the equations
$$\mu(p_i)=\varphi_i, i=1,...,N.$$
Morever, the required solution $f (x)$ defined by
$$f(x) = \min_{i = \overline{1,M}} \left\{ \frac{1}{4}|\eta_i|^2 + \left\langle \eta_i , x -\frac{1}{2} \eta_i \right\rangle \right\}.$$