Abstract:
We consider the so called Hilbert boundary value problem with boundary condition in the unit disk. Its coficient is assumed to be Hölder-continuous everywhere on the unit circle excluding a finite set of points. At these points its argument has nonremovable discontinuity of logarithmic order. We obtain formulas for the general solution and describe completely the solvability picture in a class of analytic and bounded functions in unit disc. Our technique is based on the theory of entire functions of zero-order approximation and the geometric theory of functions. The results obtained are applied to the study of the solvability of a single boundary value problem for a certain class generalized analytic function.
Keywords:Riemann–Hilbert problem, maximum principle, infinite index, entire functions of zero-order approximation, generalized analytic function.
Citation:
P. L. Shabalin, A. Kh. Fatykhov, “Inhomogeneous Hilbert boundary value problem with several points of logarithmic turbulence”, Izv. Vyssh. Uchebn. Zaved. Mat., 2021, no. 1, 64–80; Russian Math. (Iz. VUZ), 65:1 (2021), 57–71
\Bibitem{ShaFat21}
\by P.~L.~Shabalin, A.~Kh.~Fatykhov
\paper Inhomogeneous Hilbert boundary value problem with several points of logarithmic turbulence
\jour Izv. Vyssh. Uchebn. Zaved. Mat.
\yr 2021
\issue 1
\pages 64--80
\mathnet{http://mi.mathnet.ru/ivm9641}
\crossref{https://doi.org/10.26907/0021-3446-2021-1-64-80}
\transl
\jour Russian Math. (Iz. VUZ)
\yr 2021
\vol 65
\issue 1
\pages 57--71
\crossref{https://doi.org/10.3103/S1066369X21010059}
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Linking options:
https://www.mathnet.ru/eng/ivm9641
https://www.mathnet.ru/eng/ivm/y2021/i1/p64
This publication is cited in the following 4 articles:
P. L. Shabalin, R. R. Faizov, “The Hilbert problem in a half-plane for generalized analytic functions with a singular point on the real axis”, jour, 166:1 (2024), 111
A. Darya, N. Tagkhizadekh, “Zadachi Shvartsa i Dirikhle dlya ˉ∂-uravneniya v treugolnoi oblasti”, Izv. vuzov. Matem., 2024, no. 11, 12–22
Pavel Shabalin, Rafael Faizov, E. Vdovin, “Hilbert boundary value problem for generalized analytic functions with a singular line”, E3S Web Conf., 274 (2021), 11003
Nail Tuktamyshov, E. Vdovin, “Explosive technologies in transport construction”, E3S Web Conf., 274 (2021), 02002