Geodesic
Linear conjugation problem for elliptic systems in the plane
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference on Mathematical Modelling in Applied Sciences — ICMMAS'19. Belgorod, August 20–24, 2019, Tome 195 (2021), pp. 108-117.

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

In an open set $D=\mathbb{C}\setminus\Gamma$ bounded by a Lyapunov contour $\Gamma$ of class $C^{1,\nu}$, we consider the linear conjugation problem for first-order elliptic systems with constant complex and real leading coefficients. Using the integral representation of solutions by a generalized Cauchy-type integral and a generalized Pompeiu integral obtained in this paper, we reduce the original systems to equivalent systems of integral equations. Under certain conditions on the coefficients, the right-hand sides of the systems, and the right-hand side of the boundary condition, using the integral representation obtained and the results of the classical theory of singular operators, we establish a criterion for the Fredholm solvability of the problems posed obtain a formula for the index.
Keywords: weighted Hölder space, linear conjugation problem, Fredholm operator, elliptic system.
Mots-clés : index formula 
@article{INTO_2021_195_a12,
     author = {A. P. Soldatov and O. V. Chernova},
     title = {Linear conjugation problem for elliptic systems in the plane},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {108--117},
     publisher = {mathdoc},
     volume = {195},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_195_a12/}
}
TY  - JOUR
AU  - A. P. Soldatov
AU  - O. V. Chernova
TI  - Linear conjugation problem for elliptic systems in the plane
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2021
SP  - 108
EP  - 117
VL  - 195
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2021_195_a12/
LA  - ru
ID  - INTO_2021_195_a12
ER  - 
%0 Journal Article
%A A. P. Soldatov
%A O. V. Chernova
%T Linear conjugation problem for elliptic systems in the plane
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2021
%P 108-117
%V 195
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2021_195_a12/
%G ru
%F INTO_2021_195_a12
A. P. Soldatov; O. V. Chernova. Linear conjugation problem for elliptic systems in the plane. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference on Mathematical Modelling in Applied Sciences — ICMMAS'19. Belgorod, August 20–24, 2019, Tome 195 (2021), pp. 108-117. http://geodesic.mathdoc.fr/item/INTO_2021_195_a12/

[7] Abapolova E. A., Soldatov A. P., “K teorii singulyarnykh integralnykh uravnenii na gladkom konture”, Nauch. ved. BelGU. Ser. Mat. Fiz., 5 (76):18 (2010), 6–20

[8] Boyarskii B. V., “Teoriya obobschennogo analiticheskogo vektora”, Ann. Polon. Math., 17:3 (1966), 281–320

[9] Vekua I. N., Obobschennye analiticheskie funktsii, Nauka, M., 1988

[10] Gakhov F. D., Kraevye zadachi, Nauka, M., 1977

[11] Muskhelishvili N. I., Singulyarnye integralnye uravneniya, Nauka, M., 1968 | MR

[12] Soldatov A. P., “Metod teopii funktsii v kpaevykh zadachakh na ploskosti. I. Gladkii sluchai”, Izv. AH SSSR. Cep. mat., 55:5 (1991), 1070–1100

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

[14] Soldatov A. P., Chernova O. V., “Zadacha Rimana—Gilberta dlya ellipticheskikh sistem pervogo poryadka na ploskosti s postoyannymi starshimi koeffitsientami”, Itogi nauki i tekhn. Sovr. mat. prilozh. Temat. obz., 149 (2018), 95–102

[15] Gilbert R. P., Buchanan J. L., First-Order Elliptic Systems: A Function Theoretic Approach, Academic Press, New York, 1983 | MR | Zbl