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

Ricardo Estrada ()
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Ricardo Estrada: Louisiana State University

Partial Differential Equations and Applications, 2020, vol. 1, issue 1, 1-13

Abstract: 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 $$\Omega ,$$Ω, $${\mathbf {a}}$$a is a fixed point of $$\Omega ,$$Ω, and if for all homogeneous polynomials p of degree k the one dimensional function $$\begin{aligned} \varphi _{p}\left( r\right) =\int _{\mathbb {S}}u\left( \mathbf {a+} r\omega \right) p\left( \omega \right) \,\mathrm {d} \omega, \end{aligned}$$φpr=∫Sua+rωpωdω,is a polynomial of degree k at the most in some interval $$0\le r

Keywords: Harmonic functions; Harmonic polynomials; Mean value theorems; Primary 31B05, 33C55, 35B05; Secondary 46F10 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s42985-019-0003-z

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