Finding the solution of the elliptic system of the first order in the form of a vector-function

Abstract

The relationship between Douglis analytic functions and the analytic functions of the Laplace equation for elliptic equations with constant coefficients serves the same purpose. In this sense, some boundary value problems solved by Douglis using analytic functions remain relevant today. Douglis’s Bitsadze-Samarsky problem on analytic functions was reconsidered by A.P. Soldatov. Meanwhile, for certain types of matrices and regions, V.G. Nikolaev proved the existence and uniqueness of the Schwarz problem solution. In any region G of the complex plane C, we considered a first order system y J x =F, where the eigenvalues of the constant matrix J Cl l lie in the upper half-plane, Imv > 0. In the case where l = 1 J = J(z) is continuous and ImJ > 0, we obtain the Beltrami equation. In this paper, to find the solution of elliptic equation with constant coefficient, the corresponding J-analytic system was constructed by choosing the matrix J of size 2 x 2andJ has no real eigenvalues. In the postcomponential notation, the system consists of two differential equations depending on the variable z. Thus, the solution of this equation was found using the vector- function.