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Integral identities on a sphere for normal derivatives of polyharmonic functions. (Russian. English summary) Zbl 1383.31001
Summary: Identities for the integrals over the unit sphere of the products of linear combinations of normal derivatives of polyharmonic function in the unit ball and homogeneous harmonic polynomials are obtained. Basing on these identities the necessary conditions for the values on the unit sphere of polynomials on normal derivatives of polyharmonic functions are derived. Illustrative examples are given.

MSC:
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
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[1] V.V. Karachik, On the mean value property for polyharmonic functions in the ball, Siberian Advances in Mathematics, 24:3 (2014), 169-182.
[2] I.I. Bavrin, Operatory dlya garmonicheskih funkcij i ih prilozheniya, Differencial’nye uravneniya, 21:1 (1985), 9-15. MR0777774
[3] V.V. Karachik, A problem for the polyharmonic equation in the sphere, Siberian Mathematical Journal, 32:5 (1991), 767-774. MR1155803 ИНТЕГРАЛЬНЫЕ ТОЖДЕСТВА НА СФЕРЕ551 · Zbl 0777.31006
[4] B.D. Koshanov, A.P. Soldatov, Boundary value problem with normal derivatives for a higher-order elliptic equation on the plane, Differential Equations, 52:12 (2016), 1594-1609. MR3604680 · Zbl 1368.35102
[5] R. Dalmasso, On the mean-value property of polyharmonic functions, Studia Sci. Math. Hungar., 47:1 (2010), 113-117. MR2654232 · Zbl 1240.31009
[6] V.V. Karachik, On the mean-value property for polyharmonic functions, Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 6:3 (2013), 59-66. Zbl 1291.31002
[7] K.O. Besov, On the boundary behavior of components of polyharmonic functions, Math. Notes, 64:4 (1998), 450—460. MR1687236 · Zbl 0939.31007
[8] V.V. Karachik, On some special polynomials and functions, Siberian Electronic Mathematical Reports, 10 (2013), 205-226. MR3040001 · Zbl 1330.33004
[9] V.V. Karachik, Construction of polynomial solutions to some boundary value problems for Poisson’s equation, Computational Mathematics and Mathematical Physics, 51:9 (2011), 1567-1587. MR2907145
[10] V.V. Karachik, On some special polynomials, Proceedings of the American Mathematical Society, 132:4 (2004), 1049-1058. MR2045420 · Zbl 1058.33009
[11] V.V. Karachik, P -Latin matrices and Pascal’s triangle modulo a prime, Fibonacci Quarterly, 34:4 (1996), 362-372. MR1394767 · Zbl 0862.11024
[12] V.V. Karachik, Solvability conditions for the Neumann problem for the homogeneous polyharmonic equation, Differential Equations, 50:11 (2014), 1449-1456. MR3369154 · Zbl 1319.35032
[13] V.V. Karachik, On the arithmetic triangle arising from the solvability conditions for the Neumann problem, Mathematical Notes, 96:1-2 (2014), 217-227. MR3344291 · Zbl 1317.31017
[14] A.V. Bizadze, O nekotoryh svojstvah poligarmonicheskih funkcij, Differencial’nye uravneniya, 24:5 (1988), 825-831. MR0951248
[15] V.V. Karachik, On solvability conditions for the Neumann problem for a polyharmonic equation in the unit ball, Journal of Applied and Industrial Mathematics, 8:1 (2014), 63- 75. MR3234793
[16] F. Gazzola, G. Sweers, H.-Ch. Grunau, Polyharmonic boundary value problems, Lecture Notes Math., 1991, Springer, 2010.
[17] G.C. Verchota, The biharmonic Neumann problem in Lipschitz domains, Acta Math., 194 (2005), 217-279. MR2231342 · Zbl 1216.35021
[18] V.V. Karachik, Construction of polynomial solutions to the Dirichlet problem for the polyharmonic equation in a ball, Computational Mathematics and Mathematical Physics, 54:7 (2014), 1122-1143. MR3233567 · Zbl 1313.35093
[19] B. Turmetov, R. Ashurov, On Solvability of the Neumann Boundary Value Problem for Non- homogeneous Biharmonic Equation, British Journal of Mathematics & Computer Science, 4:4 (2014), 557-571.
[20] V.V. Karachik, A Neumann-type problem for the biharmonic equation, Siberian Advances in Mathematics, 27 · Zbl 1380.31004
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