Poincaré problem with measurable data for semilinear Poisson equation in the plane





Poincaré and Neumann boundary-value problems, measurable boundary data, logarithmic capacity, semilinear equations of the Poisson type, nonlinear sources, angular limits, nontangent paths


We study the Poincaré boundary-value problem with measurable in terms of the logarithmic capacity boundary data for semilinear Poisson equations defined either in the unit disk or in Jordan domains with quasihyperbolic boundary condition. The solvability theorems as well as their applications to some semilinear equations, modelling diffusion with absorption, plasma states and stationary burning, are given.


Download data is not yet available.


Gutlyanskiĭ, 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

Sobolev, S. L. (1963). Applications of functional analysis in mathematical physics. Translations of Mathematical Monographs. (Vol. 7). Providence, R. I.: AMS.

Gutlyanskiĭ, V. Ya., Nesmelova, O. V., Ryazanov, V. I. & Yefimushkin, A. S. (2022). Hilbert problem with measurable data for semilinear equations of the Vekua type. Dopov. Nac. akad. nauk Ukr., No. 2, pp. 3-10. https: //doi. org/10. 15407/dopovidi2022. 02. 003

Gutlyanskii, V., Ryazanov, V., Yakubov, E. & Yefimushkin, A. (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

Leray, J. & Schauder, Ju. (1934). Topologie et equations fonctionnelles. Ann. Sci. Ecole Norm. Sup., 51, No. 3, pp. 45-78. https: //doi. org/10. 24033/asens. 836

Ahlfors, L. (1966). Lectures on quasiconformal mappings. New York: Van Nostrand.

Diaz, J. I. (1985). Nonlinear partial differential equations and free boundaries. Vol. 1: Elliptic equations. Research Notes in Mathematics. (Vol. 106). Boston: Pitman.

Aris, R. (1975). The mathematical theory of diffusion and reaction in permeable catalysts. Vol. I, II. Oxford: Clarendon Press. https: //doi. org/10. 1007/BF02459545

Gutlyanskiĭ, 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

Zeldovic, Ya. B., Barenblatt, G. I., Librovich, V. B. & Mahviladze, G. M. (1985). The mathematical theory of combustion and explosions. New York: Consult. Bureau.

Pokhozhaev, S. I. (2010). Concerning an equation in the theory of combustion. Math. Notes, 88, No. 1-2, pp. 48-56. https: //doi. org/10. 1134/S0001434610070059




How to Cite

Gutlyanskiĭ, V. ., Nesmelova, O. ., Ryazanov, V. ., & Yefimushkin, A. . (2022). Poincaré problem with measurable data for semilinear Poisson equation in the plane. Reports of the National Academy of Sciences of Ukraine, (4), 10–18. https://doi.org/10.15407/dopovidi2022.04.010