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

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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.

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Notes

  1. For example the expression \(\omega _{1}+\omega _{2}^{3}\) does not look homogeneous, but corresponds to an element of \({\mathcal {P}}_{3},\) namely \(x_{1}r^{2} +x_{2}^{3}.\)

  2. We follow the usual convention that repeated indices are to be summed.

  3. Using the arguments of [4] we can just assume that \(u_{,i}\) are locally integrable.

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Authors and Affiliations

  1. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Ricardo Estrada

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  1. Ricardo Estrada
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Correspondence to Ricardo Estrada.

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This article is part of the section “Theory of PDEs” edited by Eduardo Teixeira.

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Estrada, R. Characterization of harmonic functions by the behavior of means at a single point. SN Partial Differ. Equ. Appl. 1, 2 (2020). https://doi.org/10.1007/s42985-019-0003-z

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