Abstract
We develop a method for the reduction of the Dirichlet problem for an axisymmetric potential in a simply connected domain of the meridian plane to a Cauchy singular integral equation. In the case where the boundary of the domain is smooth and satisfies certain additional conditions, we regularize the indicated singular integral equation.
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REFERENCES
M. A. Lavrent'ev and B. V. Shabat, Problems of Hydrodynamics and Their Mathematical Models [in Russian], Nauka, Moscow (1977).
N. Radzhabov, “Some boundary-value problems for an equation of the axisymmetric field theory,” in: Investigations on Boundary-Value Problems in the Theory of Functions and Differential Equations [in Russian], Academy of Sciences of Tadzhik SSR, Dushanbe (1965), pp. 79–128.
L. G. Mikhailov and N. Radzhabov, “An analog of the Poisson formula for second-order equations with singular line,” Dokl. Akad. Nauk Tadzh. SSR, 15, No. 11, 6–9 (1972).
N. Radzhabov, “Construction of potentials and investigation of interior and exterior boundary-value problems of the Dirichlet and Neumann type for the Euler — Poisson — Darboux equation on a plane,” Dokl. Akad. Nauk Tadzh. SSR, 17, No. 8, 7–11 (1974).
S. A. Plaksa, “Dirichlet problem for axisymmetric potential fields in a disk of the meridian plane. I,” Ukr. Mat. Zh., 52, No. 4, 492–511 (2000).
S. A. Plaksa, “On integral representations of an axisymmetric potential and the Stokes flow function in domains of the meridian plane. I,” Ukr. Mat. Zh., 53, No. 5, 631–646 (2001).
I. P. Mel'nichenko and S. A. Plaksa, “Potential fields with axial symmetry and algebras of monogenic functions of vector variables. I,” Ukr. Mat. Zh., 48, No. 11, 1518–1529 (1996).
I. P. Mel'nichenko and S. A. Plaksa, “Potential fields with axial symmetry and algebras of monogenic functions of vector variables. II,” Ukr. Mat. Zh., 48, No. 12, 1695–1703 (1996).
F. D. Gakhov, Boundary-Value Problems [in Russian], Nauka, Moscow (1977).
M. A. Lavrent'ev and B. V. Shabat, Methods of the Theory of Functions of Complex Variables [in Russian], Nauka, Moscow (1987).
N. I. Muskhelishvili, Singular Integral Equations [in Russian], Nauka, Moscow (1968).
S. A. Plaksa, “On a composition of singular and regular integrals on a rectifiable curve,” in: Contemporary Problems in Approximation Theory and Complex Analysis [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1990), pp. 104–112.
S. A. Plaksa, “On repeated integrals on a rectifiable curve,” in: Complex Analysis and Potential Theory [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1992), pp. 82–100.
A. F. Timan, Theory of Approximation of Functions of a Real Variable [in Russian], Fizmatgiz, Moscow (1960).
G. M. Goluzin, Geometric Theory of Functions of a Complex Variable [in Russian], Gostekhizdat, Moscow (1952).
A. Zygmund, “Sur le module de continuite de la somme de la serie conjuguee de la serie de Fourier,” Prace Mat.-Fiz., 33, 125–132 (1924).
L. G. Magnaradze, “On one generalization of the Privalov theorem and its application to certain boundary-value problems in the theory of functions and to singular integral equations,” Dokl. Akad. Nauk SSSR, 68, No. 4, 657–660 (1949).
Ya. L. Geronimus, “On some properties of functions continuous in a closed disk,” Dokl. Akad. Nauk SSSR, 98, No. 6, 889–891 (1954).
S. E. Warschawski, “On differentiability at the boundary in conformal mapping,” Proc. Amer. Math. Soc., 12, No. 4, 614–620 (1961).
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Plaksa, S.A. Dirichlet Problem for an Axisymmetric Potential in a Simply Connected Domain of the Meridian Plane. Ukrainian Mathematical Journal 53, 1976–1997 (2001). https://doi.org/10.1023/A:1015486805984
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DOI: https://doi.org/10.1023/A:1015486805984