Geodesic
On the Schwarz problem for the Moisil--Teodoresco system
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Mathematical Modeling, Tome 188 (2020), pp. 3-13.

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The Schwarz problem for the classical Cauchy–Riemann system in a flat domain is to determine its solution by the known first component of this solution on the boundary of the domain. In this paper, we consider an analog of this problem for the Moisil—Teodorescu system in a three-dimensional domain bounded by a smooth surface. We propose explicit expressions of the kernel and cokernel of this problem in terms of topological invariants of the domain and the kernel and cokernel of the integral representation of solutions of the Moisil—Teodorescu system that are closely related to the Schwarz problem.
Keywords: Moisil–Teodorescu system, Schwarz problem, multiply connected domain.
Mots-clés : kernel, cokernel 
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A. P. Soldatov. On the Schwarz problem for the Moisil--Teodoresco system. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Mathematical Modeling, Tome 188 (2020), pp. 3-13. http://geodesic.mathdoc.fr/item/INTO_2020_188_a0/

[1] Bitsadze A. V., “Prostranstvennyi analog integrala tipa Koshi i nekotorye ego prilozheniya”, Dokl. AN SSSR., 93:3 (1953), 389–392 | Zbl

[2] Bitsadze A. V., Kraevye zadachi dlya ellipticheskikh uravnenii vtorogo poryadka, Nauka, M., 1966

[3] Bott R., Tu L. V., Differentsialnye formy v algebraicheskoi topologii, Nauka, M., 1989

[4] Polunin V. A., Soldatov A. P., “Zadacha Rimana—Gilberta dlya sistemy Moisila—Teodoresku v konechnoi oblasti”, Neklassicheskie uravneniya matematicheskoi fiziki, IM SO RAN, Novosibirsk, 2010, 192–201

[5] Polunin V. A., Soldatov A. P., “O sopryazhennoi zadache Rimana—Gilberta dlya sistemy Moisila—Teodoresku”, Nauch. ved. Belgorod. un-ta., 5:22 (2011), 106–111

[6] Polunin V. A., Soldatov A. P., “Trekhmernyi analog integrala tipa Koshi”, Differ. uravn., 47:5 (2011), 366–375 | MR

[7] Soldatov A. P., “Singulyarnye integralnye operatory i ellipticheskie kraevye zadachi”, Sovr. mat. Fundam. napr., 63 (2017), 1–189

[8] Khatcher A., Algebraicheskaya topologiya, Izd-vo MTsNMO, M., 2011

[9] Grigor'ev Yu., “Quaternionic functions and their applications in a viscous fluid flow”, Complex Anal. Oper. Theory., 12 (2018), 491–508 | DOI | MR | Zbl

[10] Grigor'ev Yu., Gürlebeck K, Legatiuk D., “Quaternionic formulation of a Cauchy problem for the Lamé equation”, AIP Conf. Proc., 2017, 280007

[11] Gürlebeck K., Habetha K., Sprossig W., Holomorphic Functions in the Plane and $n$-dimensional Space, Birkhäuser, Basel–Boston–Berlin, 2008 | MR | Zbl

[12] Moisil G. C., Theodorescu N., “Fonction holomophic dans l'espace”, Bul. Soc. St. Cluj., 6 (1931), 177–194 | MR | Zbl

[13] Moisil G. C., Theodorescu N., “Fonctions holomorphes dans l’space”, Math. Cluj., 5 (1931), 142–159 | MR

[14] Stein E. M., Weiss G., “Generilization of the Cauchy–Riemann equation and representations of the rotation group”, Am. J. Math., 90 (1968), 163–196 | DOI | MR | Zbl

[15] Shapiro M., Vasilevski N., “Quaternionic $\Psi$-hyperholomorphic functions, singular integral operators, and boundary-value problems, I. $\Psi$-Hyperholomorphic function theory”, Compl. Var. Ellipt. Equ., 27:1 (1995), 17–46 | MR | Zbl

[16] Shapiro M., Vasilevski N., “Quaternionic $\Psi$-hyperholomorphic functions, singular integral operators, and boundary-value problems, II. Algebras of singular integral operators and Riemann-type boundary-value problems”, Compl. Var. Ellipt. Equ., 27:1 (1995), 67–96 | MR | Zbl