Green's functions and correct restrictions of the polyharmonic operator
DOI:
https://doi.org/10.26577/JMMCS.2021.v109.i1.03Keywords:
Уравнение Пуассона, полигармонические уравнения, задача Дирихле, задача Неймана, задача Робена, корректные сужения оператора.Abstract
In this paper, for completeness of presentation, we give explicitly the Green's functions for the classical problems - Dirichlet, Neumann, and Robin for the Poisson equation in a multidimensional unit ball. There are various ways of constructing the Green's function of the Dirichlet problem for the Poisson equation. For many types of areas, it is built explicitly. Recently, there has been renewed interest in the explicit construction of Green's functions for classical problems. The Green's function of the Dirichlet problem for a polyharmonic equation in a multidimensional ball is constructed in an explicit form, and for the Neumann problem the construction of the Green's function remains an open problem. The paper gives a constructive way of constructing the Green's function of Dirichlet problems for a polyharmonic equation in a multidimensional ball. Finding general well-posed boundary value problems for differential equations is always an urgent problem. In this paper, we briefly outline the theory of restriction and extension of operators and describe well-posed boundary value problems for a polyharmonic operator.
References
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[2] Vladimirov V.S., Uravneniya matematicheskoy fiziki [Equations of mathematical physics]. (Moscow: Nauka, 1981).
[3] Sobolev S.L., Vvedeniye v teoriyu kubaturnykh formul [Introduction to the theory of cubature formulas]. (Moscow: Nauka, 1974).
[4] Zorich V.A., Matematicheskiy analiz: Uchebnik. Chast’ II. [Mathematical Analysis: Textbook. Part II.] (Moscow:
Fizmatlit, 1984).
[5] Sadybekov M.A., Torebek B.T., Turmetov B.Kh., Representation of Green’s function of the Neumann problem for a multi-dimensional ball. Comp. Var. and Ell. Eq. – 2016. – 61:1. – 104–123.
[6] Sadybekov M.A., Turmetov B.Kh., Torebek B.T., On an explicit form of the Green function of the Roben problem for the Laplace operator in a circle. Adv. Pure Appl. Math. – 2015. – 6:3. 163–172.
[7] Kalmenov T.Sh., Koshanov B.D., Nemchenko M.Yu., Green function representation for the Dirichlet problem of the polyharmonic equation in a sphere. Comp. Var. and Ell. Eq. – 2008. – 53:2. 177–183.
[8] Kalmenov T.Sh., Koshanov B.D., Representation for the Green’s function of the Dirichlet problem for the polyharmonic equations in a ball. Sib. Math. Journal. – 2008. – 49:3. 423–428. Doi: 10.1007/s11202-008-0042-8
[9] Kalmenov T.Sh., Suragan D., On a new method for constructing the Green’s function of the Dirichlet problem for a polyharmonic equation. Differen. eq. – 2012. – 48:3. – 435–438. Doi: 10.1134/S0012266112030160
[10] Begehr H., Biharmonic Green functions. Le matematiche. – 2006. – 61. – 395–405.
[11] Wang Y., Ye L., Biharmonic Green function and biharmonic Neumann function in a sector. Comp. Var. and Ell. Eq. – 2013. – 58:1. – 7–22.
[12] Wang Y. Tri-harmonic boundary value problems in a sector. Comp. Var. and Ell. Eq. – 2014. – 59:5. – 732–749.
[13] Begehr H., Du J., Wang Y., A Dirichlet problem for polyharmonic functions. Ann. Mat. Pura Appl. – 2008. – 187:4. – 435–457.
[14] Begehr H., Vaitekhovich T., Harmonic boundary value problems in half disc and half ring. Funct. Approx. Comment. Math. – 2009. – 40:2. – 251–282.
[15] Begehr H., Vaitekhovich T., Modified harmonic Roben function. Comp. Var. and Ell. Eq. – 2013. – 58:4. – 483–496.
[16] Neumann J.von., Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren. Math. Ann. – 1929. – 102. – 49–131.
[17] Vishik M.I., Ob obshchikh krayevykh zadachakh dlya ellipticheskikh differentsial’nykh uravneniy [General boundary value problems for elliptic differential equations]. Trudy Mat. O-va. – 1952. – 3. – 187–246.
[18] Vishik M.I., Boundary value problems for elliptic equations degenerating on the boundary of the domain. Matem. Sbor. – 1954. – 7:3. – 1307–1311.
[19] Bitsadze A.V., Samarskyi A.A., On some simplest generalizations of linear elliptic boundary value problems. Dokl. Akad. Nauk SSSR. 1969. – 185:4. – 739–740.
[20] Dezin A.A. Partial differential equations. Berlin etc. Springer-Verlag, 1987.
[21] Kokebaev B.K., Otelbaev M., Shynybekov A.N., On the theory of restriction and extension of operators. I. Izves. AN KazSSR. Ser. fiz-mat. 1982. – 5. – 24–27.
[22] Kokebaev B.K., Otelbaev M., Shynybekov A.N., On the theory of restriction and extension of operators. II. Izves. AN KazSSR. Ser. fiz-mat. – 1983. – 1. – 24–27.
[23] Kokebaev B.K., Otelbaev M., Shynybekov A.N., On the expansion and restriction of operators. Dokl. Akad. Nauk SSSR. – 1983. – 271:6. – 1307–1311.
[24] Oynarov R., Parasidi I.N., Correctly Solvable Extensions of Operators with Finite Defects in a Banach Space. Izves. AN KazSSR. Ser. fiz-mat. – 1988. – 5. – 35–44.
[25] Burenkov V.I., Otelbaev M., On the singular numbers of correct restrictions of non-selfadjoint elliptic differential operators. Eurasian Math. J. 2011. – 2:1. – 145–148.
[26] Koshanov B.D., Otelbaev M., Correct Contractions stationary Navier-Stokes equations and boundary conditions for the setting pressure. AIP Conf. Proc. 2016. – 1759. /10.1063/1.4959619
[27] Kanguzhin B.E., Changes in a finite part of the spectrum of the Laplace operator unter delta-like perturbations. Differen. Eq. 2019. – 55:10. – 1428–1335.
[28] Kanguzhin B.E., Tulenov K.S., Singular perturbations Changes of Laplace operator and their recolvents. Comp. Var. and Ell. Eq. 2020. – 65:9. – 1433–1444.
[29] Biyarov B.N., Svistunov D.L., Abdrasheva G.K., Correct singular perturbations of the Laplace operator. Eurasian Mathematical Journal. 2020. – 11:4. – 25–34.
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How to Cite
Koshanov, B. D. (2021). Green’s functions and correct restrictions of the polyharmonic operator. Journal of Mathematics, Mechanics and Computer Science, 109(1), 35–55. https://doi.org/10.26577/JMMCS.2021.v109.i1.03
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Differential and Integral Equations
