Geodesic
On decisions of Schwartz' problem for $J$-analytic functions with the same Jordan basis of real and imaginary parts of $J$-matrix
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 3, pp. 410-422.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $l$-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.
Keywords: Schwartz' problem, $l$-holomorphic function, Lyapunov line, Jordan basis, characteristic equation, elliptic system, Dirichlet's problem.
Mots-clés : Jordan form 
@article{VSGTU_2016_20_3_a1,
     author = {V. G. Nikolaev},
     title = {On decisions of {Schwartz'} problem for $J$-analytic functions with the same {Jordan} basis of real and imaginary parts of $J$-matrix},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {410--422},
     publisher = {mathdoc},
     volume = {20},
     number = {3},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2016_20_3_a1/}
}
TY  - JOUR
AU  - V. G. Nikolaev
TI  - On decisions of Schwartz' problem for $J$-analytic functions with the same Jordan basis of real and imaginary parts of $J$-matrix
JO  - Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
PY  - 2016
SP  - 410
EP  - 422
VL  - 20
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VSGTU_2016_20_3_a1/
LA  - ru
ID  - VSGTU_2016_20_3_a1
ER  - 
%0 Journal Article
%A V. G. Nikolaev
%T On decisions of Schwartz' problem for $J$-analytic functions with the same Jordan basis of real and imaginary parts of $J$-matrix
%J Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
%D 2016
%P 410-422
%V 20
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VSGTU_2016_20_3_a1/
%G ru
%F VSGTU_2016_20_3_a1
V. G. Nikolaev. On decisions of Schwartz' problem for $J$-analytic functions with the same Jordan basis of real and imaginary parts of $J$-matrix. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 3, pp. 410-422. http://geodesic.mathdoc.fr/item/VSGTU_2016_20_3_a1/

[1] Soldatov A. P., “The Schwarz problem for Douglis analytic functions”, J. Math. Sci., 173:2 (2011), 221–224 | DOI | MR | Zbl

[2] Nikolaev V. G., Soldatov A. P., “On the solution of the Schwarz problem for $J$-analytic functions in a domain bounded by a Lyapunov contour”, Differ. Equ., 51:7 (2015), 962–966 | DOI | MR | Zbl

[3] Soldatov A. P., “Integral representation of Douglis analytic functions”, Vestnik SamGU. Estestvenno-Nauchnaya Ser., 2008, no. 8/1(67), 225–234 (In Russian) | MR

[4] Soldatov A. P., “Hyperanalytic functions and their applications”, Sovr. Mat. Pril., 15 (2004), 142–199 (In Russian) | MR

[5] Soldatov A. P., “The Hardy space of solutions to first-order elliptic systems”, Doklady Mathematics, 76:2 (2007), 660–664 | DOI | MR | Zbl

[6] Soldatov A. P., “High-order elliptic systems”, Differ. Equ., 25:1 (1989), 109–115 | MR | Zbl

[7] Bitsadze A. V., “On the uniqueness of the solution of the Dirichlet problem for elliptic partial differential equations”, Uspekhi Mat. Nauk, 3:6(28) (1948), 211–212 (In Russian) | MR | Zbl

[8] Bizadse A. W., Grundlagen der Theorie analytischer Funktionen, I. Abteilung, Band 23, Mathematische Lehrbücher und Monographien, Akademie-Verlag, Berlin, 1973, 186 pp. (In German) | MR | MR | Zbl | Zbl

[9] Bitsadze A. V., Boundary value problems for second order elliptic equations, North-Holland Series in Applied Mathematics and Mechanics, 5, North-Holland Publ. Company, Amsterdam, 1968, 211 pp. | MR | Zbl | Zbl

[10] Boyarskii B. V., “The theory of generalized analytic vector”, Annales Polonici Mathematici, 17:3 (1965), 281–320 (In Russian) https://eudml.org/doc/265098

[11] Vekua I. N., Generalized analytic functions, International Series of Monographs on Pure and Applied Mathematics, 25, Pergamon Press, Oxford, 1962, xxvi+668 pp. | MR | MR | Zbl | Zbl

[12] Zhura N. A., “Bitsadze-Samarskij type boundary-value problems for Douglis-Nirenberg elliptical systems”, Differ. Equ., 28:1 (1992), 79–88 | MR | Zbl

[13] Zhura N. A., “A general boundary value problem for systems elliptic in the Douglis- Nirenberg sense in domains with smooth boundary”, Russ. Acad. Sci., Izv., Math., 44:1 (1995), 21–42 | DOI | MR | Zbl

[14] Muskhelishvili N. I., Singular integral equations, Wolters-Noordhoff Publ., Groningen, 1967, 447 pp. | MR | Zbl | Zbl

[15] Nikolaev V. G., “On some properties of $J$-analytical functions”, Vestnik SamGU. Estestvenno-Nauchnaya Ser., 2013, no. 3(104), 25–32 (In Russian) | Zbl

[16] Nikolaev V. G., Panov E. Yu., “On coincidence of $\lambda$- and $\mu$-holomorphic functions on the boundary of a domain and applications to elliptic boundary value problems”, Problemy matematicheskogo analiza [Problems of mathematical analysis], Issue 74, eds. N. N. Ural'tseva, Tamara Rozhkovskaia, Novosibirsk, 2013, 123–132 (In Russian)