Weighted means of harmonic functions and characterization of balls

TL;DR

This paper explores weighted mean value identities for harmonic functions and their derivatives, introducing new analytic characterizations of balls based on volume mean conditions involving weights.

Contribution

It presents novel weighted mean identities for harmonic functions and derives new geometric characterizations of balls using these identities.

Findings

Weighted mean identities involve logarithmic and other weights.

New analytic characterizations of balls are established.

Applications of weighted identities are demonstrated.

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.