Weighted means of harmonic functions and characterization of balls

Abstract

Weighted mean value identities over balls are considered for harmonic functions and their derivatives. Logarithmic and other weights are involved in these identities for functions. Some applications of weighted identities are presented. Also, new analytic characterizations of balls are proved; each of them requires the volume mean of a single weight function over the domain under consideration to be equal to a prescribed number depending on the weight.