Функций Грина и корректные сужения полигармонического оператора
DOI:
https://doi.org/10.26577/JMMCS.2021.v109.i1.03Ключевые слова:
Poisson equation, polyharmonic equations, Dirichlet problem, Neumann problem, Roben problem, correct restrictions of the operatorАннотация
В данной работе приведены в явном виде функций Грина классических задач -- Дирихле, Неймана и Робена для уравнения Пуассона в многомерном единичном шаре. Существуют различные способы построения функции Грина задачи Дирихле для уравнения Пуассона. Для многих видов областей она построена в явном виде. В последнее время возобновился интерес к построению в явном виде функций Грина классических задач. Функция Грина задачи Дирихле для полигармонического уравнения в многомерном шаре построена в явном виде, а для задачи Неймана построение функции Грина остается открытой задачей. В работе дан конструктивный способ построения функции Грина задач Дирихле для полигармонического уравнения в многомерном шаре. Нахождение общих корректных краевых задач для дифференциальных уравнений всегда является актуальной задачей. В данной работе кратко изложена теория сужения и расширения операторов и описаны корректные краевые задачи для полигармонического оператора.
Библиографические ссылки
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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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Как цитировать
Koshanov, B. D. (2021). Функций Грина и корректные сужения полигармонического оператора. Вестник КазНУ. Серия математика, механика, информатика, 109(1), 35–55. https://doi.org/10.26577/JMMCS.2021.v109.i1.03
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Раздел
Дифференциальные и интегральные уравнения
