# On continuous solutions of the model homogeneous Beltrami equation with a polar singularity

### Abstract

This pap er consists of two parts. The first part is devoted to the study of the Beltrami mo delequation with a p olar singularity in a circle c entered at the ori gin , with a cut along the p ositivesemiaxis. The c o efficients of the equation have a first-order p ole at the origin and do not even b elongto the class L 2 ( G ) . For this reason, despite its specific form, this equation is not covered by theanalytical apparatus of I.N. Vekua [1] and needs to be independently studied. Using the techniquedeveloped by A.B. Tungatarov [2] in combination with the methods of the theory of functions of acomplex variable [3] and functional analysis [4], manifolds of continuous solutions of the Beltramimodel equation with a polar singularity are obtained. The theory of these equations has numerousapplications in mechanics and physics. In the second part of the article, the coefficients of theequation are chosen so that the resulting solutions are continuous in a circle without a cut [5].These results can be used in the theory of infinitesimal bendings of surfaces of positive curvaturewith a flat point and in constructing a conjugate isometric coordinate system on a surface ofpositive curvature with a planar point [6].### References

1. Vekua I.N. Generalized analytic functions. M .: Nauka, 1988. P.512.

2. Tungatarov A.B. On a method for constructing continuous solutions of the Carleman-Vekua equation with a singular point // Differential Equations. -1992. T.28, No. 8. S.1427-1434.

3. Titchmarsh E. (Titchmarsh E.C.) Theory of functions. -M .: –Science, 1980. –P.464.

4. Kolmogorov A.N., Fomin S.V. Elements of function theory and functional analysis. -M.: –Science, 1976. –P.544.

5. Usmanov Z.D. On infinitesimal bendings of surfaces of positive curvature with an isolated flat point // Mat.collection.- 1970. –V.83(125):4(12).- P.596-615.

6. Usmanov Z.D. On a class of generalized Cauchy-Riemann systems with a singular point // Sib. math. journal. -1973.-V.14. № 5, P. 1076-1087.

7. Vekua I.N. Systems of differential equations of an elliptic type and boundary problems with application to the theory of shells // Matem.sb.- 1952. –T.31 (73), No. 2. -C.234-314.

8. Vekua I.N. Fixed points of generalized analytic functions // DAN SSSR. - 1962. –T.145, No. 1. –S.24-26.

9. Vekua I.N. Fundamentals of tensor analysis and covariant theory. -M .: –Science, 1978. –P.296.

10. Bliev N.K. Generalized analytic functions in fractional spaces, –Alma-Ata. Science, [In Russian],1985, p. 159.

11. Generalized analytic functions in the fractional space, 1997, USA, Addison Wesleu Longhmah Juc. Pitman Monographs, № 86.

12. Otelbaev M.O. Ospanov K.N. On a generalized Cauchy-Riemann system with nonsmooth coefficients // Izv.vuzov.Mathematika.-1989.- No. 3.

13. Tungatarov A.B. Properties of an integral operator in classes of summable functions, Izv. SSR. Ser. Phys.-Math. 132 (5), 58–62 (1985).

14. Abreu-Blaya R., Bory-Reyes J., and Pena-Pena D. On the jump problem for β-analytic functions,Complex Variables and elliptic equat. 51 (8–11), 763–775 (2006).

15. Abreu-Blaya R., Bory-Reyes J., and Vilaire J.-M. A jump problem for β-analytic functions in domains with fractal boundaries, Revista Matem. Complutense 23, 105–111 (2010).

16. Abreu-Blaya R., Bory-Reyes J., and Vilaire J.-M. The Riemann boundary value problem for β-analytic functions over D-summable closed curves, International J. of Pure and Appl. Math. 75 (4), 441–453 (2012).

17. Usmanov Z.D. Infinitesimal bending of surfaces of positive curvature with a flat point // Differential Geometry. Banach Center Publications. Warsaw. -1984. –V.12 –P.241-272.

18. Tungatarov A.B. On a Carleman-Vekua equation with a polar singularity // Bulletin of KazGU. Ser.Mat. 1995. No. 3. -S.145-159.

19. Bliev N.K., Tungatarov A.B. On a generalized Cauchy-Riemann system with a singular point // Differential equations and their applications. –Alma-Ata. Science, 1975.

20. Kasymova D.E. The study of solutions of elliptic systems in the plane with a singular point: Abstract of thesis. ... candidate of physical and mathematical sciences. –Almaty, 1999.

21. Markushevich A.I. Selected chapters of the theory of analytic functions -Science, -M., 1976.

22. Bitsadze A.V. Fundamentals of the theory of analytic functions of a complex variable. –M .: Nauka, 1984. –P.320.

23. Katz D.B. Marcinkiewicz exponents and the jump problem for the Beltrami equation // News of Universities. Maths.2017, № 6, P. 44–51 http://kpfu.ru/science/nauchnye-izdaniya/ivrm/

24. Katz B.A. The Riemann problem on a closed Jordan curve, Izv. universities. Mat., No. 3, 68–80 (1984)

25. Drozhzhinov Yu.N., Zavialov B.I. Introduction to the theory of generalized functions. Mat. inst. them. Steklova V.A., RAS, -M.: 2006.

2. Tungatarov A.B. On a method for constructing continuous solutions of the Carleman-Vekua equation with a singular point // Differential Equations. -1992. T.28, No. 8. S.1427-1434.

3. Titchmarsh E. (Titchmarsh E.C.) Theory of functions. -M .: –Science, 1980. –P.464.

4. Kolmogorov A.N., Fomin S.V. Elements of function theory and functional analysis. -M.: –Science, 1976. –P.544.

5. Usmanov Z.D. On infinitesimal bendings of surfaces of positive curvature with an isolated flat point // Mat.collection.- 1970. –V.83(125):4(12).- P.596-615.

6. Usmanov Z.D. On a class of generalized Cauchy-Riemann systems with a singular point // Sib. math. journal. -1973.-V.14. № 5, P. 1076-1087.

7. Vekua I.N. Systems of differential equations of an elliptic type and boundary problems with application to the theory of shells // Matem.sb.- 1952. –T.31 (73), No. 2. -C.234-314.

8. Vekua I.N. Fixed points of generalized analytic functions // DAN SSSR. - 1962. –T.145, No. 1. –S.24-26.

9. Vekua I.N. Fundamentals of tensor analysis and covariant theory. -M .: –Science, 1978. –P.296.

10. Bliev N.K. Generalized analytic functions in fractional spaces, –Alma-Ata. Science, [In Russian],1985, p. 159.

11. Generalized analytic functions in the fractional space, 1997, USA, Addison Wesleu Longhmah Juc. Pitman Monographs, № 86.

12. Otelbaev M.O. Ospanov K.N. On a generalized Cauchy-Riemann system with nonsmooth coefficients // Izv.vuzov.Mathematika.-1989.- No. 3.

13. Tungatarov A.B. Properties of an integral operator in classes of summable functions, Izv. SSR. Ser. Phys.-Math. 132 (5), 58–62 (1985).

14. Abreu-Blaya R., Bory-Reyes J., and Pena-Pena D. On the jump problem for β-analytic functions,Complex Variables and elliptic equat. 51 (8–11), 763–775 (2006).

15. Abreu-Blaya R., Bory-Reyes J., and Vilaire J.-M. A jump problem for β-analytic functions in domains with fractal boundaries, Revista Matem. Complutense 23, 105–111 (2010).

16. Abreu-Blaya R., Bory-Reyes J., and Vilaire J.-M. The Riemann boundary value problem for β-analytic functions over D-summable closed curves, International J. of Pure and Appl. Math. 75 (4), 441–453 (2012).

17. Usmanov Z.D. Infinitesimal bending of surfaces of positive curvature with a flat point // Differential Geometry. Banach Center Publications. Warsaw. -1984. –V.12 –P.241-272.

18. Tungatarov A.B. On a Carleman-Vekua equation with a polar singularity // Bulletin of KazGU. Ser.Mat. 1995. No. 3. -S.145-159.

19. Bliev N.K., Tungatarov A.B. On a generalized Cauchy-Riemann system with a singular point // Differential equations and their applications. –Alma-Ata. Science, 1975.

20. Kasymova D.E. The study of solutions of elliptic systems in the plane with a singular point: Abstract of thesis. ... candidate of physical and mathematical sciences. –Almaty, 1999.

21. Markushevich A.I. Selected chapters of the theory of analytic functions -Science, -M., 1976.

22. Bitsadze A.V. Fundamentals of the theory of analytic functions of a complex variable. –M .: Nauka, 1984. –P.320.

23. Katz D.B. Marcinkiewicz exponents and the jump problem for the Beltrami equation // News of Universities. Maths.2017, № 6, P. 44–51 http://kpfu.ru/science/nauchnye-izdaniya/ivrm/

24. Katz B.A. The Riemann problem on a closed Jordan curve, Izv. universities. Mat., No. 3, 68–80 (1984)

25. Drozhzhinov Yu.N., Zavialov B.I. Introduction to the theory of generalized functions. Mat. inst. them. Steklova V.A., RAS, -M.: 2006.

Published

2020-04-05

How to Cite

KUSHERBAYEVA, Ulbyke; ABDUAKHITOVA, Gulzhan; ASSADI, Amir.
On continuous solutions of the model homogeneous Beltrami equation with a polar singularity.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 105, n. 1, p. 10-18, apr. 2020. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/700>. Date accessed: 07 june 2020.
Section

Mathematics

Keywords
Beltrami equation, equation with a polar singularity