On decisions of Schwartz' problem for J-analytic functions with the same Jordan basis of real and imaginary parts of J-matrix

Boundary Schwartz' problem for J-analytic functions was studied within this scientific work. These functions are solutions of linear complex system of partial differential equations of the first order. It was considered, that the real and imaginary parts of J-matrix are put into triangular form by means of one and the same complex transformation. The main theorem proved a criterion for eigenvalues of J-matrix. Shall this criterion be fulfilled within the complex plane within the boundaries defined by Lyapunov line, there is a decision on Schwartz' problem and it is the only one. The equal form of this criterion was found, which in many cases is more convenient for check. While proving the theorem, known facts about boundary properties of λ-holomorphic functions are applied. The proof itself is based on the method of direct and reverse reduction of Schwarz' problem to Dirichlet’s problem for real valued elliptic systems of partial differential equations of the second order. Examples of matrices are given, whereby the specified criterion is fulfilled.

Schwartz' problem, λ-holomorphic function, Lyapunov line, Jordan form, Jordan basis, characteristic equation, elliptic system, Dirichlet's problem