Klyachin A.A. On piecewise-linear almost-solutions of elliptic equations

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Klyachin Alеksеy Alеksandrovich

Doctor of Physical and Mathematical Sciences, Head of Department of Mathematical Analysis and Function Theory Volgograd State University
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Prospect Universitetsky, 100, 400062 Volgograd, Russian Federation

Abstract. In this paper we define deviation ε_Q (f ^N) of piecewise-linear almost-solution f ^N of the minimal surface equation Q[f(x)] = 0 and we get a general formula to calculate it.  Let  be tetrahedrons which have vertex P_i. We denote sides leaving from vertex P_i of the tetrahedron , j = 1, 2, ..., k(i), and let be  the side of tetrahedron opposite to vertex P_i. We set by the external normal vectors of the sides relatively of  . As f ^N is linear function in  then ∇f ^N =  ≡ const. Then the following equality holds

where  is exterior side relatively P_i. On the basis of this concept it obtained approximation equation ε_Q(f^N) = 0 or μ_iju_ij=(μ_ij,5 + μ_ij,6)u_i+1j + (μ_ij,2 + μ_ij,3)u_i-1j + (μ_ij,1 + μ_ij,2)u_ij+1 + (μ_ij,4 + μ_ij,5)u_ij-1,

and proved that the deviation ε_Q(u^N) converges to the integral of the modulus of the mean curvature of the graph of C²-smooth function u, that is

     Thus, the obtained system of nonlinear equations aproximate the minimal surface
equation.

Key words: piecewise-linear functions, almost-solution, minimal surface equation, approximation equation, deviation of piecewise-linear almost-solution.

On piecewise-linear almost-solutions of elliptic equations by Klyachin A.A. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №2 (19) 2013 pp. 18-25

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A. A. Klyachin URL: https://mp.jvolsu.com/index.php/en/component/attachments/download/150

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