Baltabek KANGUZHİN, Karlygash DOSMAGULOVA

Well-posed problems for the Laplace-Beltrami operator on a punctured two-dimensional sphere

An arbitrary point is removed from a three-dimensional Euclidean space on a two-dimensional sphere. The new well-posed solvable boundary value problems for the corresponding Laplace-Beltrami operator on the resulting punctured sphere are presented. To formulate the well-posed problems some properties of Green's function of the Laplace-Beltrami operator on a two-dimensional sphere are previously studied in detail.

Keywords:

Laplace-Beltrami operator, two-dimensional punctured sphere well-posed problems, Green's functions for elliptic equations,

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[1] Vishik M.I. On general boundary value problems for elliptic differential equations, Proceedings of the Moscow Mathematical Society, 1952, vol.1, pp.187-246.
[2] Pavlov B.S. The theory of extensions and explicitly soluble models, Russian Math. Surveys, 1987, 42:6, 127-168. DOI:10.1070/RM1987v042n06ABEH001491
[3] Kokebaev B.K., Otelbaev M., Shynybekov A.N. On the theory of restriction and extension of operators, I. Izv Akad NaukKaz SSR, Ser Fiz-Mat. 1982, 5:24.
[4] Kokebaev B.K., Otelbaev M., Shynybekov A.N. On questions of extension and restriction of operators. Dokl Akad Nauk SSSR. 1983, 271(6), pp.1307-1310.
[5] Kokebaev B.K., Otelbaev M., Shynybekov A.N. On the theory of restriction and extension of operators, II. Izv Akad NaukKaz SSR, Ser Fiz-Mat. 1983, 110(1):24.
[6] Kanguzhin B., Akanbay Y., Kaiyrbek Z. On the uniqueness of the recovery of the domain of the perturbed Laplace operator // Lobachevskii Journal of Mathematics, 2022, 43(6). pp. 1532-1535. ISSN 1995-0802 DOI: 10.1134/S1995080222090116
[7] Kanguzhin B.E., Tulenov K.S. Correctness of the definition of the Laplace operator with delta-like potentials // Complex Variables and Elliptic Equations, 2022, 67(4), pp. 898-920. DOI:10.1080/17476933.2020.1849164
[8] Kanguzhin B., Fazullin Z. On the localization of the spectrum of some perturbations of a two-dimensional harmonic oscillator // Complex Variables and Elliptic Equations, 2021, 66(6-7), pp. 1194-1208. DOI: 10.1080/17476933.2021.1885386
[9] Kanguzhin B.E., Tulenov K.S. Singular perturbations of Laplace operator and their resolvents // Complex Variables and Elliptic Equations, 2020, 65(9), pp. 1433-1444. DOI:10.1080/17476933.2019.1655551
[10] Abduakhitova G.E., Kanguzhin B.E. The correct definition of second order elliptic operators with point interactions and their resolvents // Siberian Advances in Mathematics, 2020, 30(3), pp. 153-161. DOI:10.3103/s1055134420030013
[11] Kanguzhin B.E. Changes in a finite part of the spectrum of the Laplace operator under delta-Like perturbations. Differential Equations // 2019, 55(10), pp.1328- 1335. DOI:10.1134/S0012266119100082
[12] Zhapsarbayeva L.K., Kanguzhin B.E., Konyrkulzhaeva M.N. Self-adjoint restrictions of maximal operator on graph // Ufa Mathematical Journal, 2017, 9(4), pp.35-43. DOI:10.13108/2017-9-4-35
[13] Kanguzhin B., Nurakhmetov D., Tokmagambetov N. On green function’s properties // International Journal of Mathematical Analysis, 2013, 7(13-16), pp.747-753. DOI:10.12988/ijma.2013.13073
[14] Kanguzhin B.E., Aniyarov A.A. Well-posed problems for the Laplace operator in a punctured disk. Mathematical Notes, 2011, 89(5), 819-829. DOI:10.4213/mzm8767
[15] Kanguzhin BE, Tokmagambetov NE. Resolvents of well-posed problems for finite rank perturbations of the polyharmonic operator in a punctured domain. Siberian Math J. 2016;57(2):265-273. DOI:10.17377/smzh.2016.57.209.
[16] Hobson E.W. The theory of spherical and ellipsoidal harmonics. Cambridge at the University Press. 1931.
[17] Feng D., Yuan X. Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer Monographs in Mathematics, Springer New York Heidelberg Dordrecht London. ISSN 1439-7382 ISBN 978-1-4614-6659-8 ISBN 978-1-4614-6660-4 (eBook) DOI:10.1007/978-1-4614-6660-4
[18] Abramowitz, M. and Stegun, C. A. (Eds.). ”Legendre Functions” and ”Orthogonal Polynomials.” Ch. 22 in Chs. 8 and 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 331-339 and 771-802, 1972.
[19] Gradshteyn I.S., Ryzhik I.M. Table of Integrals, Series, and Products Seventh Edition. Academic Press is an imprint of Elsevier, ISBN-13: 978-0-12-373637-6,