Volume 6, no. 3Pages 59 - 66

On the Mean-Value Property for Polyharmonic Functions

V.V. Karachik
The mean-value property for normal derivatives of polyharmonic function on the unit sphere is obtained. The value of integral over the unit sphere of normal derivative of $m$th order of polyharmonic function is expressed through the values of the Laplacian's powers of this function at the origin. In particular, it is established that the integral over the unit sphere of normal derivative of degree not less then 2k-1 of k-harmonic function is equal to zero. The values of polyharmonic function and its Laplacian's powers at the center of the unit ball are found. These values are expressed through the integral over the unit sphere of a linear combination of the normal derivatives up to k-1 degree for the k-harmonic function. Some illustrative examples are given.
Full text
polyharmonic functions, mean-value property, normal derivatives on a sphere.
1. Stein E.M., Weiss G. Introduction to Fourier Analysis on Euclidian Spaces. Princeton Univ. Press, Princeton, NJ, 1971.
2. Dalmasso R. On the Mean-Value Property of Polyharmonic Functions. Studia Sci. Math. Hungar., 2010, vol. 47, no. 1, pp. 113-117.
3. Karachik V.V. On Some Special Polynomials and Functions. Siberian Electronic Mathematical Reports, 2013, vol. 10, pp. 205-226.
4. Karachik V.V. Construction of Polynomial Solutions to Some Boundary Value Problems for Poisson's Equation. Computational Mathematics and Mathematical Physics, 2011, vol. 51, no. 9, pp. 1567-1587.
5. Karachik V.V. A Problem for the Polyharmonic Equation in the Sphere. Siberian Mathematical J., 2005, vol. 32, no. 5, pp. 767-774.
6. Karachik V.V. On One Representation of Analytic Functions by Harmonic Functions. Siberian Advances in Mathematics, 2008, vol. 18, no. 2, pp. 103-117.
7. Karachik V.V. On Some Special Polynomials. Proceedings of American Mathematical Society, 2004, vol. 132, no. 4, pp. 1049-1058.
8. Менихес Л.Д. For Regularization of Certain Classes of Mappings Inverse to Integral Operators [O regulyarizuemosti nekotorykh klassov otobrazheniy, obratnykh k integral'nym operatoram]. Matematicheskie Zametki [Mathematical Notes], 1999, vol. 65, no. 2, pp. 222-229.
9. Menikhes L.D. On Sufficient Condition for Regularizability of Linear Inverse Problems. Mathematical Notes, 2007, vol. 82, no. 1-2, pp. 242-246.