On the convergence of almost polynomial solutions of the minimal surface

In this paper we consider the polynomial approximate solutions of the Dirichlet problem for minimal surface equation. It is shown that under certain conditions on the geometric structure of the domain the absolute values of the gradients of the solutions are bounded as the degree of these polynomials increases. The obtained properties imply the uniform convergence of approximate solutions to the exact solution of the minimal surface equation. In numerical solving of boundary value problems for equations and systems of partial differential equations, a very important issue is the convergence of approximate solutions. The study of this issue is especially important for nonlinear equations since in this case there is a series of difficulties related with the impossibility of employing traditional methods and approaches used for linear equations. At present, a quite topical problem is to determine the conditions ensuring the uniform convergence of approximate solutions obtained by various methods for nonlinear equations and system of equations of variational kind. In this case, it is natural to employ variational methods of solving boundary value problems. And this is an issue on the justification of these methods arises, which is reduced to studying general properties of approximate solutions. We consider the issue on convergence of approximate solutions for the minimal surface equation
$$ \sum_{i=1}^n\frac{\partial}{\partial x_i}\left(\frac{f_{x_i}} {\sqrt{1+|\nabla f|^2}}\right)=0 $$
in domain $\Omega$ subject to the boundary condition
$$ f|_{\partial\Omega}=\varphi|_{\partial\Omega}, $$
where $\varphi\in C(\overline{\Omega})$. It should be noted that this Dirichlet problem is not solvable for an arbitrary domain (even with a smooth boundary). For planar domains the necessary and sufficient condition for the solvability of the Dirichlet problem for an arbitrary continuous boundary function $\varphi(x)$ is the convexity of this domain. In the space of dimension greater than two, such condition is the non-negativity of the mean curvature of the boundary w.r.t. the outward normal. In the present work we impose no conditions for domain, but we assume that for a given boundary function problem (1)–(2) is solvable. It is clear that for an arbitrary domain , such functions $\Omega$ exist. We study the issue on uniform convergence of polynomial approximate solutions to the minimal surface equation constructed by means of algebraic polynomials. In work [4] a similar convergence problem for piece-wise linear approximate solution to boundary value problem (1)–(2) was solved, while in work [8] there was given a description of numerical realization of finite elements methods based on piece-wise linear functions. Let us provide required definitions. In what follows we shall be interested in the issue on uniform convergence of a sequence of polynomial solutions $\varphi+v^*_N$ as $N\to\infty$. First of all we shall show that under certain conditions, the gradients of these functions are bounded by a constant independent of $N$. This property will allow us to obtain an estimate for the rate of uniform convergence to the exact solution.