Harmonic families of hypersurfaces
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.
scientific article; zbMATH DE number 6814838