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.