DIRICHLET TYPE PROBLEM FOR THE SYSTEM OF NONLINEAR BELTRAMI EQUATIONS WITH SINGULAR POINT
https://doi.org/10.55452/1998-6688-2025-22-1-239-246
Abstract
In this paper we consider a system of first order nonlinear Beltrami equations with singular point in the angular unbounded region of the complex plane. This system of equations is used in the theory of surfaces of positive infinitesimal curvature with a density point and for the construction of isometric nodal coordinates on surfaces of positive curvature with a density point. In this paper, for this system of equations we obtain sufficient condition for the solution of the Dirichlet type problem in the space of continuous functions. For this purpose, we will use the general solution of the system of corresponding linear elliptic differential equations with singular point. The proof of existence of continuous solutions of the Dirichlet problem is based on the Schauder fixed point principle.
About the Author
U. Kusherbayeva
Al-Farabi Kazakh National University
Kazakhstan
Kazakhstan
Candidate of Physical and Mathematical Sciences, Senior Lecturer
Almaty
References
1. Radon I. (1946) On boundary value problems for a logarithmic potential. UMN, vol. 1, no. 3–4(13–4), pp. 96–124. [in Russian]
2. Danilyuk I.I. (1975) Irregular boundary value problems in the plane, M.: Nauka, 296 p. [in Russian]
3. Magnaradze L.G. (1937) Basic problems of the plane theory of elasticity for contours with angular points. Dokl. AS USSR, vol. 16, no. 3, pp. 157–161. [in Russian]
4. Lopatinsky Ya.B. (1984) Theory of general boundary value problems, 316 p. [in Russian]
5. Vekua I.N. (1988) Generalised analytic functions. Moscow: Nauka, 512 p. https://urss.ru/cgi-bin/db.pl?lang=Ru&blang=ru&page=Book&id=275914 [in Russian]
6. Vekua I.N. (1952) Systems of differential equations of elliptic type and boundary value problems with application to the theory of shells. Matem. sb., vol. 31(73), no.2, pp.234–314. https://www.mathnet.ru/rus/sm5531 [in Russian]
7. Ospanov К.Н., Otelbayev М. (1989) On the generalized Cauchy-Riemann system with nonsmooth coefficients, Izv. vyuzov. Math., no. 3, pp. 48–56; Soviet Math. (Iz. VUZ), 1989, vol. 33, no. 3, pp. 75–89. [in Russian]
8. Usmanov Z.D. (1970) About infinitesimal bending of surfaces of positive curvature with an isolated point of flattening. Matem.sb., vol. 83(125):4(12), pp. 596–615. https://www.mathnet.ru/sm 3531 [in Russian]
9. Usmanov Z.D. (1973) About one class of generalised Cauchy-Riemann systems with a singular point. Sib. matem. journal., vol. 14, no. 5, p. 1076–1087. [in Russian]
10. Usmanov Z.D. Infinitesimal bending of surfaces of positive curvature with a flattening point https://bibliotekanauki.pl/articles/719678.pdf. [in Russian]
11. Abdymanapov S.A., Tungatarov A. (2005) Some classes of elliptic systems in the plane with singular coefficients. Almaty: ‘Ғylym’, 169 p. [in Russian]
12. Tungatarov A. (1991) On continuous solutions of the Karleman-Vekua equation with singular point. DAN USSR, vol. 319, no. 3, pp. 570–573. [in Russian]
13. Tungatarov A. (2013) About one class of nonlinear systems of ordinary differential equations of the first order. Proceedings of the International Scientific and Practical Conference ‘Theory of Functions, Functional Analysis and their Applications’ devoted to the 90th anniversary of the birth of member - corr. Of the Academy of Sciences of the Kazakh SSR, Doctor of Physical Sciences, Professor T.I. Amanov, 1-volume, Semey, pp. 132–136. [in Russian]
14. Akhmed-Zaki D.K., Danaev N.T., Tungatarov A. (2012) Elliptic systems in the plane with singular coefficients along lines. TWMS Journal of Pure and Applied Mathematics. Azerbaijan, vol. 3, no. 1, pp. 3–10. https://www.naturalspublishing.com/download.asp?ArtcID=17372
15. Soldatov A.P. (2017) Singular integral operators and elliptic boundary value problems. I. Sovremennr. mat. Fundam. napravlenie, vol. 63, no. 1, pp. 1–189. [in Russian]
16. Tungatarov A. (2014) Dirichlet type problem for one class of nonlinear Karleman-Vekua equations with singular point. Vestnik KazNU. Series of Mathematics, Mechanics, Informatics, vol. 80, no. 1, pp. 102–107. [in Russian]
17. Kusherbaeva U. and Altynbek S. (2024) On initial boundary value problem for Beltrami equation with polar singularity in unbounded region. Bulletin of Abay KazNPU. Series of physical and mathematical sciences, vol. 86, no. 2 (June 2024), pp. 65–73. https://doi.org/10.51889/2959-5894.2024.86.2.006
18. Gençürk Ilker. (2023) Neumann boundary value problem for the Beltrami equation in the ring domain, Turkish Journal of Mathematics, vol. 47, no. 5, Article 20. https://doi.org/10.55730/1300-0098.3449
19. Karaca B. (2020) The Dirichlet problem for complex model differential equations, Complex Variables and Elliptic Equations, vol. 65, no. 10, pp. 1748–1762. https://doi.org/10.1080/17476933.2019.1684478
20. Kusherbaeva U., Abduakhitova G. (202) On continuous solutions of the homogeneous Beltrami equation with polar singularity, Complex Variables and Elliptic Equations, 3, vol. 69, no. 5, pp. 842–848, https://doi.org/10.1080/17476933.2023.2164886
21. Lusternik L.A., Sobolev V.I. (1965) Elements of functional analysis. Moscow: Nauka, 519 p. [in Russian]
Review
For citations:
Kusherbayeva U. DIRICHLET TYPE PROBLEM FOR THE SYSTEM OF NONLINEAR BELTRAMI EQUATIONS WITH SINGULAR POINT. Herald of the Kazakh-British technical university. 2025;22(1):239-246. (In Kazakh) https://doi.org/10.55452/1998-6688-2025-22-1-239-246
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