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A characterization of polyharmonic functions. (English) Zbl 1363.31007

The author generalizes the familiar mean value characterization of harmonic functions to the case of polyharmonic functions, as follows. Let \(m\in \mathbb{N}\cup \{0\}\) and \(u\) be a continuous function on an open set \(\Omega \) in \(\mathbb{R}^{n}\). Then \(\Delta ^{m+1}u=0\) on \(\Omega \) if and only if there are continuous functions \(u_{0},u_{1},\ldots,u_{2m}\) and \(\varepsilon :\Omega \rightarrow (0,\infty)\) such that the mean value of \(u\) over the ball \(B(x,R)\) is given by \(\sum_{k=0}^{2m}u_{k}(x)R^{k}\) whenever \(x\in \Omega \) and \(0<R<\varepsilon (x)\). This characterization is then used to define polyharmonic functions on metric measure spaces.

MSC:

31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
31E05 Potential theory on fractals and metric spaces
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