A converse to the mean value theorem for biharmonic functions (Q1413624)

It is well known that the problem of the mean value of the harmonic and polyharmonic functions is very old and it is, at the same time, actual. The author's purpose is to extend the most recent results obtained for harmonic functions, to biharmonic and polyharmonic functions \((\Delta^m f=0\), \(m> 2\)). For his necessities, the results obtained in 1993 by \textit{W. Hansen} and \textit{N. Nadirashvili} are fundamental [Acta Math. 171, No. 2, 139--163 (1993; Zbl 0808.31004) and Math. Ann. 297, No. 1, 157--170 (1993; Zbl 0794.31002)]. The principal results of this paper are contained in three theorems giving the conditions under which a function \(f\), harmonic or hyperharmonic in \(\Omega\subset\mathbb{R}^n\), satisfies a mean value theorem and, conversely, the conditions under which, from a formula of mean value type, one can deduce that \(f\) is biharmonic in \(\Omega\subset\mathbb{R}^n\).