Нейман есебiнiң фредгольмдi шешiмдiлiк шартының толықтыру шартымен пара-парлығы
DOI:
https://doi.org/10.26577/JMMCS.2021.v111.i3.04Кілттік сөздер:
жоғары реттi эллиптикалық тедеулер, жалпыланған Нейман есебi, есептiң фредгольмдi шешiмдiлiгi, шекарадағы нормал туындыларАннотация
Кешендi талдау iстерi жазықтықтағы эллипстiк теңдеулер мен аралас типтегi теңдеулердi зерттеуде классикалық бағытты құрайды же қазiргi уақытта iргелi нижелер алынды. өткен ғасырдың 60-жылдарының басында эллипстiк теңдеулер мен жүйелер үшiн Дуглистiң аналитикалық функцияларын қолдануға негiзделген жаңа теориялық же функционалды тiл пайда болды. А. П. Солдатовтың же Yeh еңбектерiнде эллипстiк теңдеулер мен жүйелер теориясында Дуглис аналитикалық функциялары маңызды рөл атқаратыны белгiлi болды. Бұл функциялар классикалық Коши-Риман жүйесiн жалпылайтын бiрiншi реттi эллипстiк жүйенiң шешiмдерi болып табылады. Бұл мақалада жазықтықтағы жоғары реттi эллипстiк теңдеу үшiн Нейманның жалпыланған есебiнiң фредгольмдiк шешiлуi зерттелген. Толықтыру шартымен (Шапиро-Лопатинский шартымен) Нейманның жалпыланған есебiнiң шешiлу шартының эквиваленттiлiгi делдендi. Зерттелетiн функциялар класындағы көрсетiлген есеп индексiнiң формуласы есептеледi.
Библиографиялық сілтемелер
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[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.
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Koshanov, B. D., & Kuntuarova, A. D. (2021). Нейман есебiнiң фредгольмдi шешiмдiлiк шартының толықтыру шартымен пара-парлығы. Қазұу Хабаршысы. Математика, механика, информатика сериясы, 111(3), 39–51. https://doi.org/10.26577/JMMCS.2021.v111.i3.04
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