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

V. A. Klyachin ; M. I. Kazanin

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 1 (2017), pp. 6-12.

Résumé

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 $$ f(x) = \min_{i = \overline{1,M}} \left\{ \frac{1}{4}|\eta_i|^2 + \left\langle \eta_i , x -\frac{1}{2} \eta_i \right\rangle \right\}. $$

Détail

Mots-clés : convex polygonal surface, triangulation
Keywords: piecewise linear function, convex set, Monge–Ampere equation.

Bibliographie

[1] A.\;D. Aleksandrov, “The Dirichlet Problem for the Equation $Det||z_{ij}||=\varphi(z_1,...,z_n,z,x_1,...,x_n)$. I”, Vestn. LGU. Ser.: Matematika, mekhanika i astronomiya, 1:1 (1958), 5–24 | Zbl

[2] I.\;M. Gelfand, A.\;V. Zelevinskiy, M.\;M. Kapranov, “Discriminants of Polynomials in Several Variables and Triangulations of Newton Polyhedra”, Leningrad Mathematical Journal, 2:3 (1990), 1–62 | MR | Zbl

[3] I.\;M. Gelfand, A.\;V. Zelevinskiy, M.\;M. Kapranov, “Hypergeometric Functions and Toral Manifolds”, Functional Analysis and Its Applications, 23:2 (1989), 12–26 | MR | Zbl

[4] V.\;A. Klyachin, “Triangulation Algorithm Based on Empty Convex Set Condition”, Science Journal of Volgograd State University. Mathematics. Physics, 2015, no. 3 (28), 27–33

[5] V.\;A. Klyachin, “On the Solvability of the Discrete Analogue of the Minkowski–Alexandrov Problem”, Izv. Sarat. un-ta. Novaya ser. Ser.: Matematika. Mekhanika. Informatika, 16:3 (2016), 281–288 | MR

[6] O.\;N. Shablovskiy, “Parametric Solutions for the Monge–Ampere Equation and Gas Flow with Variable Entropy”, Vestn. Tom. gos. un-ta. Mat. i mekhanika, 2015, no. 1 (33), 105–118