Abstract
We extend classes of closed rectifiable Jordan curves and given functions in the theory of the piecewise-continuous Riemann boundary-value problem and the characteristic singular integral equation with Cauchy kernel related to this problem.
Similar content being viewed by others
References
F. D. Gakhov, Boundary-Value Problems [in Russian], Nauka, Moscow (1977).
N. I. Muskhelishvili, Singular Integral Equations [in Russian], Nauka, Moscow (1968).
A. A. Babaev and V. V. Salaev, “Boundary-value problems and singular equations on a rectifiable contour,” Mat. Zametki, 31, No. 4, 571–580 (1982).
N. V. Govorov, Riemann Boundary-Value Problem with Infinite Index [in Russian], Nauka, Moscow (1986).
E. A. Danilov, “Dependence of the number of solutions of the homogeneous Riemann problem on the contour and the modulus of the coefficient,” Dokl. Akad. Nauk SSSR, 254, No. 6, 1305–1308 (1982).
R. K. Seifullaev, “Riemann boundary-value problem on a nonsmooth open curve,” Mat. Sb., 112, No. 2, 147–161 (1980).
G. David, “Operateurs intégraux sur certaines courbes du plan complexe,” Ann. Sci. Ecole Supér., Ser. 4, 14, No. 1, 157–189 (1984).
B. A. Kats, “On an exclusive case of the Riemann problem with oscillating coefficient,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 12, 41–50 (1981).
B. Gonzalez and J. Bory, “The homogeneous Riemann boundary-value problem on rectifiable open Jordan curves,” Cienc. Mat. Havana, 9, No. 2, 3–9 (1988).
S. A. Plaksa, “Riemann boundary-value problem with oscillating coefficient and singular integral equations on a rectifiable curve,” Ukr. Mat. Zh., 41, No. 1, 116–121 (1989).
K. Kutlu, “On Riemann boundary-value problem,” An. Univ. Timi¢oara, Ser. Mat.-Inform., 38, No. 1, 89–96 (2000).
D. Pena and J. Bory, “Riemann boundary-value problem on a regular open curve,” J. Natur. Geom., 22, No. 1, 1–17 (2002).
V. V. Salaev, “Direct and inverse estimates for a singular Cauchy integral on a closed curve,” Mat. Zametki, 19, No. 3, 365–380 (1976).
S. A. Plaksa, “Riemann boundary-value problem with infinite index of logarithmic order on a spiral-form contour. I,” Ukr. Mat. Zh., 42, No. 11, 1509–1517 (1990).
O. F. Gerus, “Some estimates for moduli of smoothness of Cauchy-type integrals,” Ukr. Mat. Zh., 30, No. 5, 594–601 (1978).
S. A. Plaksa, “Semi-Noetherian operators in incomplete spaces and singular integral equations,” Dopov. Nats. Akad. Nauk Ukr., No. 12, 27–34 (2003).
Author information
Authors and Affiliations
Additional information
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 616–628, May, 2006.
Rights and permissions
About this article
Cite this article
Vasil’eva, Y.V., Plaksa, S.A. Piecewise-continuous Riemann boundary-value problem on a rectifiable curve. Ukr Math J 58, 694–708 (2006). https://doi.org/10.1007/s11253-006-0095-4
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11253-006-0095-4