The determination of a harmonic function by its sign. (English) Zbl 1165.31001

Armitage, D. H.

Real Anal. Exch. 33(2007-2008), No. 2, 275-278 (2008).

A result of Ü. Kuran [J. Lond. Math. Soc. 41, 145-152 (1966; Zbl 0138.36403)] says that, if \(h\) is a harmonic function on \(\mathbb{R}^{n}\) and \(P\not\equiv 0\) is a harmonic polynomial such that \(hP\geq 0\) outside some compact set, then \(h\) is a constant multiple of \(P\). The author generalizes this fact. Let \( (r_{j})\) be an unbounded sequence of positive numbers and let \(S(r)\) denote the sphere of centre \(0\) and radius \(r\). It is shown that, if \(h\) is harmonic on \(\mathbb{R}^{n}\) and \(P\not\equiv 0\) is a harmonic polynomial such that \(hP\geq 0\) on \(\cup _{j}S(r_{j})\), then \(h\) is a constant multiple of \(P\). The proof relies on a result about polyharmonic functions together with Kuran’s original theorem. An interesting example, based on harmonic approximation, shows that the theorem can fail if the spheres are even slightly distorted.

Reviewer: Stephen J. Gardiner (Dublin) (MR2458244)

MSC:

31B05

Harmonic, subharmonic, superharmonic functions in higher dimensions

Keywords:

harmonic function; Liouville’s theorem; polyharmonic function

Citations:

Zbl 0138.36403