Trukhlyaeva I.V. On convergence of polynomial solutions of minimal surface in domain satisfying cone condition

https://doi.org/10.15688/mpcm.jvolsu.2020.4.1

Irina V. Trukhlyaeva
Senior Lecturer, Department of Mathematical Analysis and Function Theory,
Volgograd State University
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https://orcid.org/0000-0002-8764-6132 
Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation

Abstract. In this paper we consider the polynomial approximate solutions of the 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 (see, for instance, [1]). For nonlinear equations it is first necessary to establish some a priori estimates of the derivatives of approximate solutions.In this paper, we gave a substantiation of the variational method of solving the minimal surface equation in the case of multidimensional space.We use the same approach that we used in [3] for a two-dimensional equation. Note that such a convergence was established in [3] under the condition that a certain geometric characteristic Δ(Ω) in the domain Ω, in which the solutions are considered, is positive. In particular, domain with a smooth boundary satisfied this requirement. However, this characteristic is equal to zero for a fairly wide class of domains with piecewise-smooth boundaries and sufficiently "narrow" sections at the boundary. For example, such a section of the boundary is the vertex of a cone with an angle less than π/2. In this paper, we present another approach to determining the value of Δ(Ωin terms of which it is possible to extend the results of the work [3] in domain satisfying cone condition.

Key words: minimal surface equation, uniform convergence, approximate solution.

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On convergence of polynomial solutions of minimal surface in domain satisfying cone condition by Trukhlyaeva I.V.  is licensed under a Creative Commons Attribution 4.0 International License.

Citation in EnglishMathematical Physics and Computer Simulation. Vol. 23 No. 4 2020, pp. 5-13

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