Logarithmic capacity and Riemann and Hilbert problems for generalized analytic functions
DOI:
https://doi.org/10.15407/dopovidi2020.08.011Keywords:
generalized analytic functions, Hilbert and Riemann boundary-value problems, logarithmic capacityAbstract
The study of the Dirichlet problem with arbitrary measurable boundary data for harmonic functions in the unit disk is due to the famous Luzin dissertation. Later on, the known monograph of Vekua was devoted to boundary-value problems for generalized analytic functions, but only with Hölder continuous boundary data. The present paper contains theorems on the existence of nonclassical solutions of Riemann and Hilbert problems for generalized analy tic functions with sources whose boundary data are measurable with respect to the logarithmic capacity. Our ap proach is based on the geometric interpretation of boundary values in comparison with the classical operator approach in PDE. On this basis, one can derive the corresponding existence theorems for the Poincaré problem on directional derivatives to the Poisson equations and, in particular, for the Neumann problem with arbitrary boundary data that are measurable with respect to the logarithmic capacity. These results can be also applied to semilinear equations of mathematical physics in anisotropic inhomogeneous media.
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Vekua, I. N. (1962). Generalized analytic functions. Oxford, London, New York, Paris: Pergamon Press.
Luzin, N. N. (1915). Integral and trigonometric series. (Unpublished Doctor thesis). Moscow University, Moscow, Russia (in Russian).
Luzin, N. N. (1951). Integral and trigonometric series. Editing and commentary by Bari, N. K. & Men’shov, D.E. Moscow, Leningrad: Gostehteoretizdat (in Russian).
Gutlyanskii, V., Nesmelova, O. & Ryazanov, V. (2019). To the theory of semilinear equations in the plane. J. Math. Sci., 242, No. 6, pp. 833-859. https://doi.org/10.1007/s10958-019-04519-z
Efimushkin, A. S. & Ryazanov, V. I. (2015). On the Riemann-Hilbert problem for the Beltrami equations in quasidisks. J. Math. Sci., 211, No. 5, pp. 646-659. https://doi.org/10.1007/s10958-015-2621-0
Yefimushkin, A. & Ryazanov, V. (2016) On the Riemann-Hilbert problem for the Beltrami equations. In Complex analysis and dynamical systems VI. Part 2 (pp. 299-316). Contemporary Mathematics, 667. Israel Math. Conf. Proc. Providence, RI: Amer. Math. Soc. https://doi.org/10.5186/aasfm.2020.4552
Gutlyanskii, V. Ya., Ryazanov, V. I., Yakubov, E. & Yefimushkin, A. S. (2019). On the Hilbert problem for analytic functions in quasihyperbolic domains. Dopov. Nac. akad. nauk Ukr., No. 2, pp. 23-30. https://doi.org/10.15407/dopovidi2019.02.023
Gehring, F. W. & Martio, O. (1985). Lipschitz classes and quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A. I. Math., 10, pp. 203-219. https://doi.org/10.5186/aasfm.1985.1022
Gutlyanskii, V. Ya., Ryazanov, V. I., Yakubov, E. & Yefimushkin, A. S. (2019). On boundary-value problems in domains without (A)-condition. Dopov. Nac. akad. nauk Ukr., No. 3, pp. 17-24. https://doi.org/10.15407/dopovidi2019.03.017
Gutlyanskii, V., Ryazanov, V., Yakubov, E. & Yefimushkin, A. (2020). On Hilbert boundary value problem for Beltrami equation. Ann. Acad. Sci. Fenn. Math., 45, pp. 957-973. https://doi.org/10.5186/aasfm.2020.4552
Bagemihl, F. & Seidel, W. (1955). Regular functions with prescribed measurable boundary values almost everywhere. Proc. Natl. Acad. Sci. U.S.A., 41, pp. 740-743. https://doi.org/10.1073/pnas.41.10.740
Gutlyanskii, V., Ryazanov, V. & Yefimushkin, A. (2016). On the boundary-value problems for quasiconformal functions in the plane. J. Math. Sci., 214, No. 2, pp. 200-219. https://doi.org/10.1007/s10958-016-2769-2
Federer, H. (1969). Geometric Measure Theory. Berlin: Springer.
Krasnosel’skii, M. A., Zabreyko, P. P., Pustyl’nik, E. I. & Sobolevski, P. E. (1976). Integral operators in spaces of summable functions. Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis. Leiden: Noordhoff International Publishing.
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