Harmonic families of hypersurfaces
type: Item
instance of: scholarly article
title: Harmonic families of hypersurfaces
zbMATH Open document ID: 1382.31004
author: Eleutherius Symeonidis
published in: Complex Analysis and Operator Theory
publication date: 2017-12-01T00:00:00Z
review text: The paper deals with mean value properties for harmonic functions in subsets of \(\mathbb{R}^n\), with \(n \geq 3\). As is well known, if \(h\) is a harmonic function in a set \(\Omega\), if \(0\in \Omega\), and if \(r\) is such that the ball of radius \(r\) and center \(0\) is contained in \(\Omega\), then we have the mean value property \[ \frac{1}{\omega_n}\int_{S^{n-1}}h(rx)d\sigma(x)=h(0)\, , \] where \(\sigma\) is the Euclidean measure on the sphere \(S^{n-1}\equiv \{x \in \mathbb{R}^n: |x|=1\}\). The equality above implies, in particular, that the term on the left hand side does not depend on \(r\). Such an invariance property holds also for other families of hypersurfaces, such as, for example, ellipsoids or the revolution \[ \frac{x_1^2}{\cosh^2 r}+\frac{x_2^2+\dots+x_n^2}{\sinh^2 r}=1\, . \] As a consequence, the mean value property can be understood in a more general setting as an invariance property. Thus, in the present paper, the author investigates families of hypersurfaces for which the mean value property holds.
reviewed by: Paolo Musolino
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 31B35
zbMATH DE Number: 6814838
zbMATH Keywords: harmonic functions
zbMATH Keywords: mean value properties
MaRDI profile type: Publication
full work available at URL: s11785-017-0671-7
OpenAlex ID: W2604976616
cites work: A mean value property of harmonic functions on the interior of a hyperbola
cites work: A Poisson Integral Formula for the Ellipse and Some Applications
cites work: Q5167053
cites work: Harmonic deformation of planar curves
cites work: Q5450597
DOI: 10.1007/S11785-017-0671-7
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description: scientific article; zbMATH DE number 6814838