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 Eurasian Math. J., 2014, Volume 5, Number 4, Pages 6–24 (Mi emj171)

Schwarz problem for first order elliptic systems in unbounded sectors

M. Akelab, H. Begehrc

a Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsaa, 31982, P.O. Box 380, Saudi Arabia
b Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt
c Mathematical Institute, Free University Berlin, Arnimallee 3, D-14195 Berlin, Germany

Abstract: In this article we deal with a Schwarz-type boundary value problem for both the inhomogeneous Cauchy–Riemann equation and the generalized Beltrami equation on an unbounded sector with angle $\vartheta=\pi/n$, $n\in\mathbb N$. By the method of plane parquetingreflection and the Cauchy–Pompeiu formula for the sector, the Schwarz–Poisson integral formula is obtained. We also investigate the boundary behaviour and the $C^\alpha$-property of a Schwarz-type as well as of a Pompeiu-type operator. The solution to the Schwarz problem of the Cauchy–Riemann equation is explicitly expressed. Sufficient conditions on the coefficients of the generalized Beltrami equation are obtained under which the corresponding system of integral equations is contractive. This proves the existence of a unique solution to the Schwarz problem of the generalized Beltrami equation.

Keywords and phrases: Cauchy–Pompeiu formula, Schwarz–Poisson formula, Cauchy–Riemann equation, generalized Beltrami equation, Schwarz problem, contractive mapping principle.

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MSC: 30E25, 30G30, 45E05
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Citation: M. Akel, H. Begehr, “Schwarz problem for first order elliptic systems in unbounded sectors”, Eurasian Math. J., 5:4 (2014), 6–24

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\Bibitem{AkeBeg14} \by M.~Akel, H.~Begehr \paper Schwarz problem for first order elliptic systems in unbounded sectors \jour Eurasian Math. J. \yr 2014 \vol 5 \issue 4 \pages 6--24 \mathnet{http://mi.mathnet.ru/emj171} 

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This publication is cited in the following articles:
1. M. Ku, Y. Wang, F. He, U. Kahler, “Riemann–Hilbert problems for monogenic functions on upper half ball of $\mathbb{R}^4$”, Adv. Appl. Clifford Algebr., 27:3, SI (2017), 2493–2508
2. M. Akel, F. Alabbad, “Riemann–Hilbert-type boundary value problems on a half hexagon”, Acta. Math. Sin.-English Ser., 33:9 (2017), 1249–1266
3. M. Akel, “Riemann–Hilbert problem for the Cauchy-Riemann operator in lens and lune”, Complex Var. Elliptic Equ., 62:10, SI (2017), 1570–1588
4. M. Akel, S. R. Mondal, “Dirichlet problems in lens and lune”, Bull. Malays. Math. Sci. Soc., 41:2 (2018), 1029–1043
5. M. Akel, M. Aldawsari, “Neumann boundary value problems in fan-shaped domains”, Turk. J. Math., 42:4 (2018), 1571–1589
6. U. Aksoy, H. Begehr, A. O. Celebi, “Schwarz problem for higher-order complex partial differential equations in the upper half plane”, Math. Nachr., 292:6 (2019), 1183–1193
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