Hilbert problem with measurable data for semilinear equations of the Vekua type





Hilbert boundary-value problem, measurable boundary data, logarithmic capacity, semilinear equations of the Vekua type, nonlinear sources, angular limits, nontangent paths


We prove the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the nonlinear equations of the Vekua’s type ∂z f (z ) = h (z )q(f (z )). The found solutions differ from the classical ones, because our approach is based on the notion of boundary values in the sense of angular limits along nontangential paths. The results obtained can be applied to the establishment of existence theorems for the Poincaré and Neumann boundary-value problems for the nonlinear Poisson equations of the form ΔU (z )=H (z )Q(U (z )) with arbitrary measurable boundary data with respect to the logarithmic capacity. They can be also applied to the study of some semilinear equations of mathematical physics modeling such processes as the diffusion with absorption, plasma states, stationary burning etc. in anisotropic and inhomogeneous media.


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Gutlyanskii, V., Nesmelova, O., Ryazanov, V. & Yefimushkin A. (2021). Logarithmic potential and generalized analytic functions. J. Math. Sci., 256, pp. 735-752. https://doi.org/10.1007/s10958-021-05457-5

Gutlyanskii, V. Ya., Nesmelova, O. V., Ryazanov, V. I. & Yefimushkin, A. S. (2022). Dirichlet problem

with measurable data for semilinear equations in the plane. Dopov. Nac. akad. nauk Ukr., No. 1, рр. 11-19.


Dunford, N. & Schwartz, J. T. (1958). Linear operators. Part I. General theory. Pure and Applied Mathematics., Vol. 7. New York, London: Interscience Publishers.

Leray, J. & Schauder, Ju. (1934). Topologie et équations fonctionnelles. Ann. Sci. Ecole Norm. Sup., Ser. 3, 51, pp. 45-78. https://doi.org/10.24033/asens.836

Becker, J. & Pommerenke, Ch. (1982). Hölder continuity of conformal mappings and non-quasiconformal

Jordan curves. Comment. Math. Helv., 57, No. 2, pp. 221-225. https://doi.org/10.1007/BF02565858

Gutlyanskii, V., Nesmelova, O. & Ryazanov, V. (2018). On quasiconformal maps and semilinear equations in

the plane. J. Math. Sci., 229, No. 1, pp. 7-29. https://doi.org/10.1007/s10958-018-3659-6

Gutlyanskii, V., Nesmelova, O. & Ryazanov, V. (2020). On a quasilinear Poisson equation in the plane. Anal.

Math. Phys., 10, No. 1. https://doi.org/10.1007/s13324-019-00345-3

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

Vekua, I. N. (1962). Generalized analytic functions. Oxford, New York: Pergamon Press.

Gutlyanskii, V. Ya., Ryazanov, V. I., Yakubov, E. & Yefimushkin, A. S. (2020). On Hilbert boundary value

problem for Beltrami equation. Ann. Acad. Sci. Fenn. Math., 45, No. 2, pp. 957-973. https://doi.org/10.5186/aasfm.2020.4552

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

Goluzin, G. M. (1969). Geometric theory of functions of a complex variable. Translations of Mathematical Monographs, Vol. 26. Providence, R.I.: American Mathematical Society. https://doi.org/10.1090/mmono/026

Ahlfors, L. (1966). Lectures on quasiconformal mappings. Princeton, New Jersey, Toronto, New York, London: D. Van Nostrand Company, Inc. https://doi.org/10.1090/ulect/038

Koosis, P. (1998). Introduction to Hp spaces. Cambridge Tracts in Mathematics, vol. 115. Cambridge: Cambridge Univ. Press.




How to Cite

Gutlyanskiĭ, V. ., Nesmelova, O. ., Ryazanov, V. ., & Yefimushkin, A. . (2022). Hilbert problem with measurable data for semilinear equations of the Vekua type. Reports of the National Academy of Sciences of Ukraine, (2), 3–11. https://doi.org/10.15407/dopovidi2022.02.003