Dirichlet problem with measurable data for semilinear equations in the plane

Authors

DOI:

https://doi.org/10.15407/dopovidi2022.01.011

Keywords:

logarithmic capacity, quasilinear Poisson equations, nonlinear sources, Dirichlet problem, measurable boundary data, angular limits, nontangent paths

Abstract

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk  goes to the known dissertation of Luzin. His result was formulated in terms of angular limits (along nontangent paths) that are a traditional tool for the research of the boundary behavior in the geometric function theory. With a view to further developments of the theory of boundary-value problems for semilinear equations, the present paper is devoted to the Dirichlet problem with arbitrary measurable (over logarithmic capacity) boundary data for quasilinear Poisson equations in such Jordan domains. For this purpose, it is firstly constructed completely continuous operators generating nonclassical solutions of the Dirichlet boundary-value problem with arbitrary measurable data for the Poisson equations ΔU =G over the sources G ∈Lp, p >1. The latter makes it possible to apply the Leray— Schauder approach to the proof of theorems on the existence of regular nonclassical solutions of the measurable Dirichlet problem for quasilinear Poisson equations of the form ΔU (z ) =H (z )⋅Q(U (z )) for multipliers H ∈Lp with p >1 and continuous functions Q: R→R with Q(t ) /t →0 as t →∞. These results can be applied to some specific quasilinear equations of mathematical physics, arising under a modeling of various physical processes such as the diffusion with absorption, plasma states, stationary burning, etc. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media.

Downloads

Download data is not yet available.

References

Luzin, N. N. (1951). Integral and trigonometric series. Bari, N. K. & Men’shov, D. E. (Eds. and comment. ). Moscow, Leningrad: Gostehteoretizdat (in Russian).

Gutlyanskiĭ, 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

Gutlyanskiĭ, V., Nesmelova, O. & Ryazanov, V. (2020). On a quasilinear Poisson equation in the plane. Anal. Math. Phys., 10, No. 1, Art. 6, 14 pp. https: //doi. org/10. 1007/s13324-019-00345-3

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

Ryazanov, V. (2019). On the theory of the boundary behavior of conjugate harmonic functions. Complex Anal. Oper. Th., 13, No. 6, pp. 2899-2915. https: //doi. org/10. 1007/s11785-018-0861-y

Dunford, N. & Schwartz, J. T. (1958). Linear operators. I. General theory. New York, London: Interscience Publishers.

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 équations fonctionnelles. Ann. Sci. École Norm. Supér., 51, No. 3, pp. 45-78 (in French). https: //doi. org/10. 24033/asens. 836; (1946). Topology and functional equations. Uspehi Mat. Nauk, 1, No. 3-4, pp. 71-95 (in Russian).

Gehring, F. W. & Martio, O. (1985). Lipschitz classes and quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A. I Math., 10, pp. 203-219.

Gutlyanskii, V., Nesmelova, O. & Ryazanov, V. (2020). Semi-linear equations and quasiconformal mappings. Complex Var. Elliptic Equ., 65, No. 5, P. 823-843. https: //doi. org/10. 1080/17476933. 2019. 1631288

Ladyzhenskaya, O. A. & Ural’tseva, N. N. (1968). Linear and quasilinear elliptic equations. New York: Academic Press.

Goluzin, G. M. (1969). Geometric theory of functions of a complex variable. Translations of Mathematical Monographs, Vol. 26. Providence, R. I.: American Mathematical Society.

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

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

Gutlyanskii, V., Nesmelova, O. & Ryazanov, V. (2017). 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

Downloads

Published

30.03.2022

How to Cite

Gutlyanskiĭ, V., Nesmelova, O. ., Ryazanov, V., & Yefimushkin, A. (2022). Dirichlet problem with measurable data for semilinear equations in the plane. Reports of the National Academy of Sciences of Ukraine, (1), 11–19. https://doi.org/10.15407/dopovidi2022.01.011