Abstract
The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk 𝔻 is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with Hölder continuous data for generalized analytic functions, i.e., continuous complex-valued functions f(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form \( {\partial}_{\overline{z}}f+ af+b\overline{f}=c, \) where the complexvalued functions a; b, and c are assumed to belong to the class Lp with some p > 2 in smooth enough domains D in ℂ.
Our last paper [12] contained theorems on the existence of nonclassical solutions of the Hilbert boundaryvalue problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions f : D → ℂ such that \( {\partial}_{\overline{z}}f=g \) with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincaré problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G ∈ Lp; p > 2, with arbitrary measurable boundary data over logarithmic capacity.
The present paper is a natural continuation of the article [12] and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type \( {\partial}_{\overline{z}}f(z)=h(z)q\left(f(z)\right). \) On this basis, existence theorems were also established for the Poincar´e boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form △U(z) = H(z)Q(U(z)) with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too.
Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory.
As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.
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References
L. Ahlfors. Lectures on Quasiconformal Mappings. Van Nostrand, New York, 1966.
R. Aris. The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts. V. I–II. Oxford, Clarendon Press, 1975.
G. I. Barenblatt, Ja. B. Zel’dovic, V. B. Librovich, and G. M. Mahviladze. The mathematical theory of combustion and explosions. New York, Consult. Bureau, 1985.
J. Becker and Ch. Pommerenke. “Hölder continuity of conformal mappings and nonquasiconformal Jordan curves,” Comment. Math. Helv., 57(2), 221–225 (1982).
N. Dunford and J. T. Schwartz. Linear Operators. I. General Theory, Pure and Applied Mathematics, 7. New York, London, Interscience Publishers, 1958.
J. I. Diaz. Nonlinear partial differential equations and free boundaries. Vol. I. Elliptic equations. Research Notes in Mathematics, 106. Boston, Pitman, 1985.
A. S. Efimushkin and V. I. Ryazanov. “On the Riemann-Hilbert problem for the Beltrami equations in quasidisks,” Ukr. Mat. Bull., 12(2), 190–209 (2015); transl. in J. Math. Sci., 211(5), 646–659 (2015).
G. M. Goluzin. Geometric theory of functions of a complex variable. Transl. of Math. Monographs, 26. American Mathematical Society, Providence, R.I., 1969.
V. Gutlyanskii, O. Nesmelova, and V. Ryazanov. “On quasiconformal maps and semi-linear equations in the plane,” Ukr. Mat. Bull., 14(2), 161–19 1(2017); transl. in J. Math. Sci., 229(1), 7–29 (2018).
V. Gutlyanskii, O. Nesmelova, and V. Ryazanov. “To the theory of semi-linear equations in the plane,” Ukr. Mat. Bull., 16(1), 105–140 (2019); transl. in J. Math. Sci., 242(6), 833–859 (2019).
V. Gutlyanskii, O. Nesmelova, and V. Ryazanov. “On a quasilinear Poisson equation in the plane,” Anal. Math. Phys., 10(1), Paper No. 6, 1–14 (2020).
V. Gutlyanskii, O. Nesmelova, V. Ryazanov, and A. Yefimushkin. “Logarithmic potential and generalized analytic functions,” Ukr. Mat. Bull., 18(1), 12–35 (2021); transl. in J. Math. Sci., 256(6), 9–26 (2021).
V. Ya. Gutlyanskii, V. I. Ryazanov, E. Yakubov, and A. S. Yefimushkin. “On Hilbert boundary value problem for Beltrami equation,” Ann. Acad. Sci. Fenn. Math., 45(2), 957–973 (2020).
P. Koosis. Introduction to Hp spaces. Cambridge Tracts in Mathematics. 115. Cambridge Univ. Press, Cambridge, 1998.
J. Leray and Ju. Schauder. “Topologie et equations fonctionnelles,” Ann. Sci. Ecole Norm. Sup., 51(3), 45–78 (1934) [in French]; transl. in “Topology and functional equations,” Uspehi Matem. Nauk, 1(3–4) (13–14), 71–95 (1946).
N. N. Luzin. “On the main theorem of integral calculus,” Mat. Sb., 28, 266–294 (1912).
N. N. Luzin. Integral and trigonometric series. Dissertation. Moscow [in Russian], 1915.
N. N. Luzin. Integral and trigonometric series [in Russian]. Editing and commentary by N.K. Bari and D.E. Men’shov. Gosudarstv. Izdat. Tehn.-Teor. Lit. Moscow–Leningrad, 1951.
N. Luzin. “Sur la notion de l’integrale,” Annali Mat. Pura e Appl., 26(3), 77–129 (1917).
S. I. Pokhozhaev. “On an equation of combustion theory,” Mat. Zametki, 88(1), 53–62 (2010); transl. in Math. Notes, 88(1–2), 48–56 (2010).
V. Ryazanov. “On the Theory of the Boundary Behavior of Conjugate Harmonic Functions,” Complex Analysis and Operator Theory, 13(6), 2899–2915 (2019).
S. Saks.. Theory of the integral. Warsaw, Dover Publications Inc., New York, 1964.
S. L. Sobolev. Applications of functional analysis in mathematical physics. Transl. of Math. Mon., 7. AMS, Providence, R.I., 1963.
I. N. Vekua. Generalized analytic functions. Pergamon Press. London–Paris–Frankfurt; Addison–Wesley Publishing Co., Inc., Reading, Mass., 1962.
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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 18, No. 3, pp. 359–388, July–September, 2021.
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Gutlyanskiĭ, V., Nesmelova, O., Ryazanov, V. et al. On boundary-value problems for semi-linear equations in the plane. J Math Sci 259, 53–74 (2021). https://doi.org/10.1007/s10958-021-05604-y
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DOI: https://doi.org/10.1007/s10958-021-05604-y