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

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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 
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     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/}
}
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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/