Quarterly of Applied Mathematics

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Hilbert problem for a multiply connected circular domain and the analysis of the Hall effect in a plate

Authors: Y. A. Antipov and V. V. Silvestrov
Journal: Quart. Appl. Math. 68 (2010), 563-590
MSC (2000): Primary 30E25, 32N15; Secondary 74F15
Published electronically: June 4, 2010
MathSciNet review: 2676977
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Abstract: In this paper we analyze the Hilbert boundary-value problem of the theory of analytic functions for an $ (N+1)$-connected circular domain. An exact series-form solution has already been derived for the case of continuous coefficients. Motivated by the study of the Hall effect in a multiply connected plate we extend these results by examining the case of discontinuous coefficients. The Hilbert problem maps into the Riemann-Hilbert problem for symmetric piece-wise meromorphic functions invariant with respect to a symmetric Schottky group. The solution to this problem is derived in terms of two analogues of the Cauchy kernel, quasiautomorphic and quasimultiplicative kernels. The former kernel is known for any symmetric Schottky group. We prove the existence theorem for the second (quasimultiplicative) kernel for any Schottky group (its series representation is known for the first class groups only). We also show that the use of an automorphic kernel requires the solution to the associated real analogue of the Jacobi inversion problem, which can be bypassed if we employ the quasiautomorphic and quasimultiplicative kernels. We apply this theory to a model steady-state problem on the motion of charged electrons in a plate with $ N+1$ circular holes with electrodes and dielectrics on the walls when the conductor is placed at a right angle to the applied magnetic field.

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  • 1. Construction of the Schwarz operator by the symmetry method, Trudy Sem. Kraev. Zadačam 4 (1967), 3–10 (Russian). MR 0291425
  • 2. I.A. Aleksandrov and A.S. Sorokin, The problem of Schwarz for multiply connected circular domains, Siberian Math. J. 13 (1973), 671-692.
  • 3. Y. A. Antipov and V. V. Silvestrov, Electromagnetic scattering from an anisotropic impedance half-plane at oblique incidence: the exact solution, Quart. J. Mech. Appl. Math. 59 (2006), no. 2, 211–251. MR 2219885, 10.1093/qjmam/hbj004
  • 4. Y. A. Antipov and V. V. Silvestrov, Method of automorphic functions in the study of flow around a stack of porous cylinders, Quart. J. Mech. Appl. Math. 60 (2007), no. 3, 337–366. MR 2347761, 10.1093/qjmam/hbm010
  • 5. Y. A. Antipov and V. V. Silvestrov, Circular map for supercavitating flow in a multiply connected domain, Quart. J. Mech. Appl. Math. 62 (2009), no. 2, 167–200. MR 2496027, 10.1093/qjmam/hbp003
  • 6. Y. A. Antipov and D. G. Crowdy, Riemann-Hilbert problem for automorphic functions and the Schottky-Klein prime function, Complex Anal. Oper. Theory 1 (2007), no. 3, 317–334. MR 2336026, 10.1007/s11785-007-0020-3
  • 7. W. Burnside, On a class of automorphic functions, Proc. London Math. Soc. 23 (1892), 49-88.
  • 8. L. I. Čibrikova and V. V. Sil′vestrov, On the question of the effectiveness of the solution of Riemann’s boundary value problem for automorphic functions, Izv. Vyssh. Uchebn. Zaved. Mat. 12 (1978), 117–121 (Russian). MR 529717
  • 9. Darren Crowdy, The Schwarz-Christoffel mapping to bounded multiply connected polygonal domains, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2061, 2653–2678. MR 2165505, 10.1098/rspa.2005.1480
  • 10. T. K. DeLillo, A. R. Elcrat, and J. A. Pfaltzgraff, Schwarz-Christoffel mapping of multiply connected domains, J. Anal. Math. 94 (2004), 17–47. MR 2124453, 10.1007/BF02789040
  • 11. Tobin A. Driscoll and Lloyd N. Trefethen, Schwarz-Christoffel mapping, Cambridge Monographs on Applied and Computational Mathematics, vol. 8, Cambridge University Press, Cambridge, 2002. MR 1908657
  • 12. Y.P. Emets, Boundary Value Problems of Electrodynamics for Anisotropically Conducting Media, Naukova Dumka, Kiev, 1987.
  • 13. Hershel M. Farkas and Irwin Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York-Berlin, 1980. MR 583745
  • 14. L.R. Ford, Automorphic Functions, AMS Chelsea Publishing, Providence, RI, 2004.
  • 15. F. D. Gakhov, Boundary value problems, Dover Publications, Inc., New York, 1990. Translated from the Russian; Reprint of the 1966 translation. MR 1106850
  • 16. J. Haeusler, Exakte Lösungen von Potentialproblemen beim Halleffekt durch konforme Abbildung, Solid-State Electronics 9 (1966), 417-441.
  • 17. J. Haeusler and H.J. Lippmann, Hallgeneratoren mit kleinem Linearisierungsfehler, Solid-State Electronics 11 (1968), 173-182.
  • 18. Adolf Hurwitz, Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen, Herausgegeben und ergänzt durch einen Abschnitt über geometrische Funktionentheorie von R. Courant. Mit einem Anhang von H. Röhrl. Vierte vermehrte und verbesserte Auflage. Die Grundlehren der Mathematischen Wissenschaften, Band 3, Springer-Verlag, Berlin-New York, 1964 (German). MR 0173749
  • 19. Irwin Kra, Automorphic forms and Kleinian groups, W. A. Benjamin, Inc., Reading, Mass., 1972. Mathematics Lecture Note Series. MR 0357775
  • 20. V.V. Silvestrov, The Riemann boundary value problem for symmetric automorphic functions and its application, Theory of functions of a complex variable and boundary value problems, 93-107, Chuvash. Gos. Univ., Cheboksary, 1982.
  • 21. Lloyd N. Trefethen and Ruth J. Williams, Conformal mapping solution of Laplace’s equation on a polygon with oblique derivative boundary conditions, J. Comput. Appl. Math. 14 (1986), no. 1-2, 227–249. Special issue on numerical conformal mapping. MR 829041, 10.1016/0377-0427(86)90141-X
  • 22. I. N. Vekua, Generalized analytic functions, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962. MR 0150320
  • 23. W. Versnel, Analysis of Hall-plate shaped Van der Pauw structures, Solid-State Electronics 23 (1980), 557-563.
  • 24. W. Versnel, Analysis of a circular Hall plate with equal finite contacts, Solid-State Electronics 24 (1981), 63-68.
  • 25. R.V. Wick, Solution of the field problem of the germanium gyrator, J. Appl. Physics 25 (1954), 741-756.
  • 26. È. I. Zverovič, Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces, Uspehi Mat. Nauk 26 (1971), no. 1(157), 113–179 (Russian). MR 0409841

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Additional Information

Y. A. Antipov
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Email: antipov@math.lsu.edu

V. V. Silvestrov
Affiliation: Department of Mathematics, Gubkin Russian State University of Oil and Gas, Moscow 119991, Russia

DOI: http://dx.doi.org/10.1090/S0033-569X-2010-01189-1
Keywords: Riemann-Hilbert problem, automorphic functions, Schottky group, Hall effect
Received by editor(s): December 4, 2008
Published electronically: June 4, 2010
Additional Notes: The first author was supported in part by NSF Grant DMS0707724.
The second author was supported in part by Russian Foundation for Basic Research Grant 07-01-00038.
Article copyright: © Copyright 2010 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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