EQUIVALENCE OF THE FREDHOLM SOLVABILITY CONDITION FOR THE NEUMANN PROBLEM TO THE COMPLEMENTARITY CONDITION
DOI:
https://doi.org/10.26577/JMMCS.2021.v111.i3.04Keywords:
higher order elliptic equations, generalized Neumann problem, Fredholm solvability of the problem, normal derivatives on the boundaryAbstract
The methods of complex analysis constitute the classical direction in the study of elliptic equations and mixed-type equations on the plane and fundamental results have now been obtained. In the early 60s of the last century, a new theoretical-functional approach was developed for elliptic equations and systems based on the use of functions analytic by Douglis. In the works of A.P. Soldatov and Yeh, it turned out that in the theory of elliptic equations and systems, Douglis analytic functions play an important role. These functions are solutions of a first-order elliptic system generalizing the classical Cauchy-Riemann system. In this paper, the Fredholm solvability of the generalized Neumann problem for a high-order elliptic equation on a plane is investigated. The equivalence of the solvability condition of the generalized Neumann problem with the complementarity condition (Shapiro-Lopatinsky condition) is proved. The formula for the index of the specified problem in the class of functions under study is calculated.
References
[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.
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Koshanov, B. D., & Kuntuarova, A. D. (2021). EQUIVALENCE OF THE FREDHOLM SOLVABILITY CONDITION FOR THE NEUMANN PROBLEM TO THE COMPLEMENTARITY CONDITION. Journal of Mathematics, Mechanics and Computer Science, 111(3), 39–51. https://doi.org/10.26577/JMMCS.2021.v111.i3.04
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