The Riemann boundary-value problem is solved for the classes of open rectifiable Jordan curves extended as compared with previous results and functions defined on these curves.
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).
R. K. Seifullaev, “Riemann boundary-value problem on a nonsmooth open curve,” Mat. Sb., 112, No. 2, 147–161 (1980).
N. V. Govorov, Riemann Boundary-Value Problem with Infinite Index [in Russian], Nauka, Moscow (1986).
B. A. Kats, “On the exclusive case of the Riemann problem with oscillating coefficient,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 12, 41–50 (1981).
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, 264, No. 6, 1305–1308 (1982).
B. A. Kats, “Riemann problem on an open Jordan curve,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 12, 30–38 (1983).
B. Gonzalez and J. Reyes, “The homogeneous Riemann boundary-value problem on rectifiable open Jordan curves,” Cience. 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); English translation: Ukr. Math. J., 41, No. 1, 107–112 (1989).
B. A. Kats, “Riemann boundary-value problem on fractal arcs and arcs of infinite length. I,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 6, 7–16 (1993); English translation: Russian Math., 37, No. 6, 4–13 (1993).
B. A. Kats, “Riemann boundary-value problem on fractal arcs and arcs of infinite length. II,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 7, 17–23 (1993); English translation: Russian Math., 37, No. 7, 15–21 (1993).
K. Kutlu, “On Riemann boundary-value problem,” An. Univ. Timişoara. Ser. Mat.-Inform., 38, No. 1, 89–96 (2000).
D. Pena and J. Reyes, “Riemann boundary value problem on a regular open curve,” J. Natur. Geom., 22, No. 1, 1–17 (2002).
Yu. V. Vasil’eva and S. A. Plaksa, “Piecewise-continuous Riemann boundary-value problem on a rectifiable curve,” Ukr. Mat. Zh., 58, No. 5, 616–628 (2006); English translation: Ukr. Math. J., 58, No. 5., 694–708 (2006).
V. V. Salaev, “Direct and inverse estimates for a singular Cauchy integral along 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 contour. I,” Ukr. Mat. Zh., 42, No. 11, 1509–1517 (1990); English translation: Ukr. Math. J., 42, No. 11, 1351–1358 (1990).
O. F. Gerus, “Some estimates of smoothness moduli for the Cauchy-type integrals,” Ukr. Mat. Zh., 30, No. 5, 594–601 (1978); English translation: Ukr. Math. J., 30, No. 5, 445–460 (1978).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 11, pp. 1511–1522, November, 2010.
Rights and permissions
About this article
Cite this article
Plaksa, S.A., Kud’yavina, Y... Riemann boundary-value problem on an open rectifiable jordan curve. I. Ukr Math J 62, 1752–1765 (2011). https://doi.org/10.1007/s11253-011-0465-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-011-0465-4