Geodesic
On the mean-value property for polyharmonic functions
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 6 (2013) no. 3, pp. 59-66

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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.
Keywords: polyharmonic functions, mean-value property, normal derivatives on a sphere. 
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V. V. Karachik. On the mean-value property for polyharmonic functions. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 6 (2013) no. 3, pp. 59-66. http://geodesic.mathdoc.fr/item/VYURU_2013_6_3_a5/