Abstract
Green’s functions for the Navier and Riquier–Neumann problems for the biharmonic equation in the unit ball are constructed, and integral representations of the solution of these problems are given.
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REFERENCES
Begehr, H., Biharmonic Green functions, Le Matematiche, 2006, vol. 61, no. 2, pp. 395–405.
Begehr, H. and Vaitekhovich, T., Modified harmonic Robin function, Complex Var. Elliptic Equat., 2013, vol. 58, no. 4, pp. 483–496.
Sadybekov, M.A., Torebek, B.T., and Turmetov, B.Kh., On an explicit form of the Green function of the Robin problem for the Laplace operator in a circle, Adv. Pure Appl. Math., 2015, vol. 6, no. 3, pp. 163–172.
Ying Wang and Liuqing, Ye., Biharmonic Green function and biharmonic Neumann function in a sector, Complex Var. Elliptic Equat., 2013, vol. 58, no. 1, pp. 7–22.
Ying, Wang., Tri-harmonic boundary value problems in a sector, Complex Var. Elliptic Equat., 2014, vol. 59, no. 5, pp. 732–749.
Boggio, T., Sulle funzioni di Green d’ordine \(m \), Palermo Rend., 1905, vol. 20, pp. 97–135.
Kalmenov, T.Sh., Koshanov, B.D., and Nemchenko, M.Y., Green function representation for the Dirichlet problem of the polyharmonic equation in a sphere, Complex Var. Elliptic Equat., 2008, vol. 53, pp. 177–183.
Karachik, V.V. and Antropova, N.A., Polynomial solutions of the Dirichlet problem for the biharmonic equation in the ball, Differ. Equations, 2013, vol. 49, no. 2, pp. 251–256.
Karachik, V.V. and Turmetov, B.Kh., On Green’s function of the Robin problem for the Poisson equation, Adv. Pure Appl. Math., 2019, vol. 10, no. 3, pp. 203–214.
Karachik, V.V., The Green function of the Dirichlet problem for the triharmonic equation in the ball, Math. Notes, 2020, vol. 107, no. 1, pp. 105–120.
Karachik, V.V. and Torebek, B.T., On the Dirichlet–Riquier problem for biharmonic equations, Math. Notes, 2017, vol. 102, no. 1, pp. 31–42.
Karachik, V.V., A Neumann-type problem for the biharmonic equation, Sib. Adv. Math., 2016, vol. 27, no. 2, pp. 103–118.
Soldatov, A.P., On the Fredholm property and index of the generalized Neumann problem, Differ. Equations, 2020, vol. 56, no. 2, pp. 212–220.
Bitsadze, A.V., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1982.
Karachik, V.V., Green’s function of Dirichlet problem for biharmonic equation in the ball, Complex Var. Elliptic Equat., 2019, vol. 64, no. 9, pp. 1500–1521.
Sweers, G., A survey on boundary conditions for the biharmonic, Complex Var. Elliptic Equat., 2009, vol. 54, pp. 79–93.
Karachik, V.V., Riquier–Neumann problem for the polyharmonic equation in a ball, Differ. Equations, 2018, vol. 54, no. 5, pp. 648–657.
Vladimirov, V.S., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1981.
Alimov, Sh.A., On one problem with oblique derivative, Differ. Uravn., 1981, vol. 17, no. 10, pp. 1738–1751.
Karachik, V.V. and Turmetov, B.Kh., On the Green’s function for the third boundary value problem, Sib. Adv. Math., 2018, vol. 21, no. 1, pp. 32–43.
Karachik, V.V., On one set of orthogonal harmonic polynomials, Proc. Am. Math. Soc., 1998, vol. 126, no. 12, pp. 3513–3519.
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This work was supported by the Government of the Russian Federation, Resolution no. 211 of March 16, 2013, Agreement no. 02.A03.21.0011.
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Karachik, V.V. Green’s Functions of the Navier and Riquier–Neumann Problems for the Biharmonic Equation in the Ball. Diff Equat 57, 654–668 (2021). https://doi.org/10.1134/S0012266121050098
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DOI: https://doi.org/10.1134/S0012266121050098