Explicit solution of a class of Riemann-Hilbert problems

Darren Crowdy


Analytical solutions to a special class of Riemann-Hilbert boundary value problems on multiply connected domains are presented. The solutions are written, up to a finite number of accessory parameters, as non-singular indefinite integrals whose integrands are expressed in terms of the Schottky-Klein prime function associated with the Schottky double of the planar domain.


Riemann-Hilbert problem, boundary value problem, Schottky-Klein prime function

Mathematics Subject Classification

2000 Mathematics Subject Classification: 30C20, 30E25, 35F15


Akhiezer, N.I. "Elements of the theory of elliptical functions." AMS Translations of Mathematical Monographs. Providence, RI: 1990.

Antipov, Y. and V.V. Silvestrov. "Method of Riemann surfaces in the study of supercavitating flow around two hydrofoils in a channel." Physica D. 235 (2007): 72-81.

Baker, H.F. "Abelian functions: Abel’s theorem and the allied theory of theta functions." Cambridge: Cambridge University Press, 1897.

Beardon, A.F. "A primer on Riemann surfaces." London Mathematical Society Lecture Note Series 78. Cambridge: Cambridge University Press, 1984.

Crowdy, D.G. "The Schwarz-Christoffel mapping to bounded multiply connected polygonal domains." Proc. Roy. Soc. A. 461 (2005): 2653-2678.

---. "Schwarz--Christoffel mappings to unbounded multiply connected polygonal regions." Math. Proc. Camb. Phil. Soc. 142 (2007): 319-339.

Crowdy, D.G. and J.S. Marshall. "Conformal mappings between canonical multiply connected domains." Comput. Methods Funct. Theory 6 (2006): 59-76.

---. "Computing the Schottky-Klein prime function on the Schottky double of planar domains." Comput. Methods Funct. Theory 7 (2007): 293-308.

Gakhov, F.D. "Boundary value problems." Dover, New York: Dover Publications, 1990.

Goluzin, G.M. "Geometric theory of functions of a complex variable." Providence, RI: Amer. Math. Soc., 1969.

Gurevich, M.I. "Theory of jets in ideal fluids." New York: Academic Press, 1965.

Hejhal, D.A. "Theta functions, kernel functions and Abelian integrals." Mem. Amer. Math. Soc. 129 (1972).

Mityushev, V.V. and S.V. Rogosin. "Constructive methods for linear and nonlinear boundary value problems." Monographs & Surveys in Pure & Applied Mathematics. Chapman & all/CRC, 1999.

Mityushev, V.V. "Solution of the Hilbert boundary value problem for a multiply connected domain." Slup. Prace Mat. Przyr. Mat. Fiz. 9A (1994): 37-69.

Nehari, Z. "Conformal mapping." Dover, New York: Dover Publications, 1952.

Schiffer, M. "Recent advances in the theory of conformal mappings." appendix to: R. Courant, Dirichlet’s principle, conformal mapping and minimal surfaces. Pure and Applied Mathematics. A Series of Text and Monographs III. New York: Interscience Publishers, 1950.

Tanveer, S. "New solutions for steady bubbles in a Hele-Shaw cell." Phys. Fluids 30 (1987): 651-658.

Vekua, I.N. "Generalized analytic functions." New York: Pergamon Press, 1962.

Wegmann, R. and M.M.S. Nasser. "The Riemann-Hilbert problem and the generalizedNeumann kernel on multiply connected regions." J. Comput. Appl. Math. 214 (2008): 36-57.

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