ON CORRECT RESTRICTIONS OF A POLYHARMONIC OPERATOR
Özet
The need to study boundary value problems for elliptic equations is dictated by numerous practical applications in the theoretical study of the processes of hydrodynamics, electrostatics, mechanics, heat conduction, elasticity theory, quantum physics. The distributions of the potential of the electrostatic field are described using the Poisson equation, and when studying the vibrations of thin plates of small deflections, biharmonic equations arise.
This work is devoted to the study of the polyharmonic equation, including the construction of the Green's function for the classical Dirichlet problem for the polyharmonic equation in a multidimensional ball and the description of well-posed boundary value problems for the polyharmonic operator.
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. And for the Neumann problem in multidimensional domains, the construction of the Green's function is an open problem.
Finding general well-posed boundary value problems for differential equations is always an urgent problem. In the early 80s of the last century, Academician of the National Academy of Sciences of the Republic of Kazakhstan M. Otelbaev and his students constructed an abstract theory that allows one to describe all correct restrictions of some maximal operator and separately all correct extensions of some minimal operator, independently of each other, in terms of the inverse operator.
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 in a multidimensional ball.
Referanslar
Соболев С.Л. Введение в теорию кубатурных формул. - М.: Наука. 1974. - 808 с.
Sadybekov M.A., Torebek B.T., Turmetov B.Kh. Representation of Green's function of the Neumann problem for a multi-dimensional ball // Complex Variables and Elliptic Equations. 2016. V. 61, № 1. - P. 104-123.
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. V. 3, № 6. - P. 163-172.
Kalmenov T.Sh., Koshanov B.D., Nemchenko M.Y. Green function representation for the Dirichlet problem of the polyharmonic equation in a sphere // Complex Variables and Elliptic Equations. 2008. V. 2, № 53. - P. 177-183. Doi: http://dx.doi.org/10.1080/17476930701671726
Kalmenov T.Sh., Koshanov B.D. Representation for the Green’s function of the Dirichlet problem for the polyharmonic equations in a ball // Siberian Mathematical Journal. 2008. V. 3, № 49. - P. 423-428. Doi: http://dx.doi.org/10.1007/s11202-008-0042-8
Kalmenov T.Sh., Suragan D. On a new method for constructing the Green's function of the Dirichlet problem for a polyharmonic equation // Differential Equations. 2012. Т. 48, №3. - P. 435-438. Doi: http://dx.doi.org/10.1134/S0012266112030160
Begehr H. Biharmonic Green functions // Le matematiche. 2006. № 61. - P. 395-405.
Wang Y., Ye L. Biharmonic Green function and biharmonic Neumann function in a sector // Complex Variables and Elliptic Equations. 2013. V. 1, № 58. - P. 7-22.
Wang Y. Tri-harmonic boundary value problems in a sector // Complex Variables and Elliptic Equations. 2014. V. 5, № 59. - P. 732-749.
Begehr H., Vu T.N.H., Z.-X. Zhang. Polyharmonic Dirichlet Problems // Proceedings of the Steklov Institute of Math. 2006. №255. – P. 13-34.
Begehr H., Du J., Wang Y. A Dirichlet problem for polyharmonic functions // Ann. Mat. Pura Appl. 2008. №187(4). – C. 435-457.
Begehr H., Vaitekhovich T. Harmonic boundary value problems in half disc and half ring // Funct. Approx. Comment. Math. 2009. №40(2). – P. 251-282.
Begehr H., Vaitekhovich T. Modified harmonic Roben function // Complex Variables and Elliptic Equations. 2013. V. 4, № 58. – P. 483-496.
J.von Neumann. Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren // Math. Ann. 1929. V. 102. - P. 49-131.
Вишик М.И. Об общих краевых задачах для эллиптических дифференциальных уравнений // Труды Матем. о-ва. 1952. № 3. - C. 187-246.
Вишик М.И. Краевые задачи для эллиптических уравнений, вырождающихся на границе области // Матем. сб. 1954. №35(77). - С. 1307-1311.
Бицадзе А.В., Самарский А.А. О некоторых простейших обобщениях линейных эллиптических краевых задач // Доклады АН СССР. 1969. Т. 185, № 4. - C. 739-740.
Dezin A.A. Partial differential equations. - Berlin etc.: Springer-Verlag, 1987.
Кокебаев Б.К., Отелбаев М., Шыныбеков А.Н. К теории сужения и расширения операторов I // Известие АН КазССР. Сер. физ-мат. 1982. № 5. - C. 24-27.
Кокебаев Б.К., Отелбаев М., Шыныбеков А.Н. К теории сужения и расширения операторов II // Известие АН КазССР. Сер. физ-мат. 1983. № 1. - C. 24-27.
Отелбаев М., Кокебаев Б.К., Шыныбеков А.Н. К вопросам расширения и сужения операторов // Доклады АН СССР. - 1983. Т. 6, № 271. - C. 1307-1311.
Ойнаров Р., Парасиди И.Н. Корректно разрешимые расширения операторов с конечными деффектами в банаховом пространстве // Известие АН КазССР. Сер. физ-мат. 1988. № 5. - C. 35-44.
Koshanov B.D., Otelbaev M.O. Correct Contractions stationary Navier-Stokes equations and boundary conditions for the setting pressure // AIP Conference Proceedings. 2016. №1759. V. 1759, 020005, http://dx.doi.org/10.1063/1.4959619
Kanguzhin B.E. Changes in a finite part of the spectrum of the Laplace operator unter delta-like perturbations // Differential Equations. 2019. V. 10, № 55. - P. 1428-1335.
Kanguzhin B.E., Tulenov K.S. Singular perturbations Changes of Laplace operator and their recolvents // Complex Variables and Elliptic Equations. 2020. V. 9, № 65. - P. 1433-1444.
Biyarov B.N., Svistunov D.L., Abdrasheva G.K. Correct singular perturbations of the Laplace operator // Eurasian Mathematical Journal. 2020. V. 4, № 11. - P. 25-34.
