Эквивалентность условии фредгольмовой разрешимости задачи Неймана с условием дополнительности
DOI:
https://doi.org/10.26577/JMMCS.2021.v111.i3.04Ключевые слова:
эллиптические уравнения высокого порядка, обобщенная задача Неймана, фредгольмова разрешимость задачи, нормальные производные на границеАннотация
Методы комплексного анализа составляют классическое направление в исследовании эллиптических уравнений и уравнений смешанного типа на плоскости и в настоящее время получены фундаментальные результаты. В начале 60-х годов прошлого столетия для эллиптических уравнений и систем был развить новый теоретико-функциональный подход, основанный на использовании функций, аналитических по Дуглису. В работах А.П. Солдатова, и Yeh выяснилось, что в теории эллиптических уравнений и систем важную роль играют функции, аналитические по Дуглису. Эти функции являются решениями эллиптической системы первого порядка, обобщающей классическую систему Коши-Римана. В данной статье исследована фредгольмовая разрешимость обобщенной задачи Неймана для эллиптического уравнения высокого порядка на плоскости. Доказана эквивалентность условии разрешимости обобщенной задачи Неймана с условием дополнительности (условием Шапиро-Лопатинского). Вычислена формула для индекса указанной задачи в исследуемой классе функций.
Библиографические ссылки
[1] Bitsadze A.V., On some properties of polyharmonic functions, Differ. Equations, 24, No. 5 (1988), 825-831.
[2] Dezin A.A., The second boundary problem for the polyharmonic equation in the space W m 2, Doklady Akad. Nauk SSSR (N.S.), 96 (1954), 901-903.
[3] Malakhova N.A., Soldatov A.P. On a Boundary Value Problem for a Higher-Order Elliptic Equation, Differential
Equations, 44:8 (2008), 1111-1118. https://doi.org/10.1134/S0012266108080089.
[4] Soldatov A.P., A boundary value problem for higher order elliptic equations in many connected domain on the plane //Vladikavkazskii Matematicheskii Zhurnal, 19:3, 2017, 51-58.
[5] Koshanov B.D., Soldatov A.P., Boundary value problem with normal derivatives for a higher order elliptic eguation on the plane, Differential Equations, 52:12 (2016), 1594-1609. https://doi.org/10.1134/S0012266116120077.
[6] Kalmenov T.Sh., Koshanov B.D., Nemchenko M.Y. 2008. Green function representation for the Dirichlet problem of the polyharmonic equation in a sphere. Complex Variables and Elliptic Equations. 53(2):177-183. Doi:
10.1080/17476930701671726
[7] Kalmenov T.Sh., Koshanov B.D., Nemchenko M.Yu. 2008. Green function representation in the Dirichlet problem for polyharmonic equations in a ball. Doklady Mathematics. 78(1):528-530. Doi: 10.1134/S1064562408040169.
[8] Sadybekov M.A., Turmetov B.Kh., Torebek B.T. 2015. On an explicit form of the Green function of the Roben problem for the Laplace operator in a circle. Adv. Pure Appl. Math. 6(3):163-172.
[9] Sadybekov M.A., Torebek B.T., Turmetov B.Kh. 2016. Representation of Green’s function of the Neumann problem for a multi-dimensional ball. Complex Variables and Elliptic Equations. 61(1):104-123.
[10] Begehr H., Vu T.N.H., Zhang Z.-X. 2006. Polyharmonic Dirichlet Problems. Proceedings of the Steklov Institute of Math. 255:13-34.
[11] Begehr H., Du J., Wang Y. 2008. A Dirichlet problem for polyharmonic functions. Ann. Mat. Pura Appl. 187(4):435-457.
[12] Begehr H., Vaitekhovich T. 2013. Modefied harmonic Robin function. Complex Variables and Elliptic Equations. 58(4):483-496.
[13] Koshanov B.D. Green’s functions and correct restrictions of the polyharmonic operator. Вестник КазНУ. Серия математика, механика, информатика. 2021. 109(1):34-54. Doi: https://doi.org/10.26577/JMMCS.2021.v109.i1.03
[14] Douglis A.A., On uniqueness in Cauchy probblems for ellipilc systems of equations// Comm Pure Appl. Math. 1960, 13, No.4, P. 593-607.
[15] Agmon S., Douglis A., Nirenberg L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Commun. Pure Appl. Math., 12:4 (1959), 623-727.
[16] Gilbert R.P., Function theoretic methods in partial differential equations Academic Press, New York, 1969.
[17] Yeh R.Z. Hyperholomorphic functions and higher order partial differentials equations in the plane// Pacific Journ. of Mathem, 1990, 142, No2, P. 379-399.
[18] Soldatov A.P., Higher-order elliptic systems, Differential Equations, 25:1 (1989), 109-115.
[19] Soldatov A.P., On the Theory of Anisotropic Flat Elasticity, Journal of Mathematical Sciences, Volume 235, Issue 4, 1, P. 484-535.
[20] Soldatov A.P., Generalized potentials of double layer in plane theory of elasticity, Evrasian mathematical journal, 2014, V.5, No 4, P. 78-125.
[21] Nazarov S., Plamenevsky B.A. Elliptic problems in domains with piecewise smooth boundaries. Walter de Gruyter, 2011.
[22] Lopatinskii Ya.B., On a method of reducing boundary-value problems for a system of differential equations of elliptic type to regular integral equations, Ukrain. Math. J., 5 (1953), 123-151.
[23] Sсheсhter M. Genеral boundary value problems for elliptic partial differential equations, Comm. Purе and Appl. mathem., 1950, 12, 467-480.
[2] Dezin A.A., The second boundary problem for the polyharmonic equation in the space W m 2, Doklady Akad. Nauk SSSR (N.S.), 96 (1954), 901-903.
[3] Malakhova N.A., Soldatov A.P. On a Boundary Value Problem for a Higher-Order Elliptic Equation, Differential
Equations, 44:8 (2008), 1111-1118. https://doi.org/10.1134/S0012266108080089.
[4] Soldatov A.P., A boundary value problem for higher order elliptic equations in many connected domain on the plane //Vladikavkazskii Matematicheskii Zhurnal, 19:3, 2017, 51-58.
[5] Koshanov B.D., Soldatov A.P., Boundary value problem with normal derivatives for a higher order elliptic eguation on the plane, Differential Equations, 52:12 (2016), 1594-1609. https://doi.org/10.1134/S0012266116120077.
[6] Kalmenov T.Sh., Koshanov B.D., Nemchenko M.Y. 2008. Green function representation for the Dirichlet problem of the polyharmonic equation in a sphere. Complex Variables and Elliptic Equations. 53(2):177-183. Doi:
10.1080/17476930701671726
[7] Kalmenov T.Sh., Koshanov B.D., Nemchenko M.Yu. 2008. Green function representation in the Dirichlet problem for polyharmonic equations in a ball. Doklady Mathematics. 78(1):528-530. Doi: 10.1134/S1064562408040169.
[8] Sadybekov M.A., Turmetov B.Kh., Torebek B.T. 2015. On an explicit form of the Green function of the Roben problem for the Laplace operator in a circle. Adv. Pure Appl. Math. 6(3):163-172.
[9] Sadybekov M.A., Torebek B.T., Turmetov B.Kh. 2016. Representation of Green’s function of the Neumann problem for a multi-dimensional ball. Complex Variables and Elliptic Equations. 61(1):104-123.
[10] Begehr H., Vu T.N.H., Zhang Z.-X. 2006. Polyharmonic Dirichlet Problems. Proceedings of the Steklov Institute of Math. 255:13-34.
[11] Begehr H., Du J., Wang Y. 2008. A Dirichlet problem for polyharmonic functions. Ann. Mat. Pura Appl. 187(4):435-457.
[12] Begehr H., Vaitekhovich T. 2013. Modefied harmonic Robin function. Complex Variables and Elliptic Equations. 58(4):483-496.
[13] Koshanov B.D. Green’s functions and correct restrictions of the polyharmonic operator. Вестник КазНУ. Серия математика, механика, информатика. 2021. 109(1):34-54. Doi: https://doi.org/10.26577/JMMCS.2021.v109.i1.03
[14] Douglis A.A., On uniqueness in Cauchy probblems for ellipilc systems of equations// Comm Pure Appl. Math. 1960, 13, No.4, P. 593-607.
[15] Agmon S., Douglis A., Nirenberg L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Commun. Pure Appl. Math., 12:4 (1959), 623-727.
[16] Gilbert R.P., Function theoretic methods in partial differential equations Academic Press, New York, 1969.
[17] Yeh R.Z. Hyperholomorphic functions and higher order partial differentials equations in the plane// Pacific Journ. of Mathem, 1990, 142, No2, P. 379-399.
[18] Soldatov A.P., Higher-order elliptic systems, Differential Equations, 25:1 (1989), 109-115.
[19] Soldatov A.P., On the Theory of Anisotropic Flat Elasticity, Journal of Mathematical Sciences, Volume 235, Issue 4, 1, P. 484-535.
[20] Soldatov A.P., Generalized potentials of double layer in plane theory of elasticity, Evrasian mathematical journal, 2014, V.5, No 4, P. 78-125.
[21] Nazarov S., Plamenevsky B.A. Elliptic problems in domains with piecewise smooth boundaries. Walter de Gruyter, 2011.
[22] Lopatinskii Ya.B., On a method of reducing boundary-value problems for a system of differential equations of elliptic type to regular integral equations, Ukrain. Math. J., 5 (1953), 123-151.
[23] Sсheсhter M. Genеral boundary value problems for elliptic partial differential equations, Comm. Purе and Appl. mathem., 1950, 12, 467-480.
Загрузки
Опубликован
2021-10-09
Выпуск
Раздел
Математика
