# Characterization of harmonic functions by the behavior of means at a single point

Original Paper
Part of the following topical collections:
1. Theory of PDEs

## Abstract

We give a characterization of harmonic functions by a mean value type property at a single point. We show that if u is real analytic in $$\Omega ,$$ $${\mathbf {a}}$$ is a fixed point of $$\Omega ,$$ and if for all homogeneous polynomials p of degree k the one dimensional function
\begin{aligned} \varphi _{p}\left( r\right) =\int _{\mathbb {S}}u\left( \mathbf {a+} r\omega \right) p\left( \omega \right) \,\mathrm {d} \omega, \end{aligned}
is a polynomial of degree k at the most in some interval $$0\le r<\eta _{p},$$ then u is harmonic in $$\Omega .$$ If u is smooth, and $$\eta _{p}=\eta$$ does not depend on p,  then we show that u must be harmonic in the ball of center $${\mathbf {a}}$$ and radius $$\eta .$$ We also give a result that applies to distributions. Furthermore, we characterize harmonic functions by flow integrals around a single point.

## Keywords

Harmonic functions Harmonic polynomials Mean value theorems

## Mathematics Subject Classification

Primary 31B05 33C55 35B05; Secondary 46F10

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