A Taylor series condition for harmonic extensions. (English) Zbl 1055.31003

Coffman, Adam; Legg, David; Pan, Yifei

Real Anal. Exch. 28(2002-2003), No. 1, 235-253 (2003).

Let \(u\) be harmonic on some open ball in \(\mathbb{R}^n\) centred at the origin, and let \(x= (x',x_n)\) denote a typical point of \(\mathbb{R}^n= \mathbb{R}^{n-1}\times \mathbb{R}\). Suppose that the Taylor series of \(u(x',0)\) and \((\partial u/\partial x_n)(x',0)\) about \(0'\) converge when \(| x'|< r\). Then it is shown that the Taylor series of \(u(x)\) converges when \(| x'|+| x_n|< r\). The proof uses elementary arguments.

Reviewer: Stephen J. Gardiner (Dublin)

MSC:

31B05	Harmonic, subharmonic, superharmonic functions in higher dimensions
35C10	Series solutions to PDEs

Keywords:

harmonic function; Taylor series