Integral representation of harmonic functions defined outside a compact set in a harmonic space
type: Item
instance of: scholarly article
title: Integral representation of harmonic functions defined outside a compact set in a harmonic space
zbMATH Open document ID: 1050.31009
DOI: 10.1023/A:1020571803677
published in: Potential Analysis
publication date: 2003-04-27T00:00:00Z
review text: The classical Bôcher theorem in \(\mathbb{R}^n\), after a Kelvin transformation, can be stated as follows: Let \(h\geq 0\) be harmonic outside a compact set in \(\mathbb{R}^n\), \(n\geq 2\). Then \(h\) is of the form \[ h(x): \begin{cases} \alpha\log| x|+ b(x),\quad &\text{if }n=2,\\ \alpha| x|^{2-n}+ b(x),\quad &\text{if }n\geq 3,\end{cases} \] where \(\alpha\) is a constant and \(b(x)\) is bounded harmonic outside a compact set. The authors prove a similar result in the context of the (Brelot) axiomatic potential theory: Let \(\Omega\) be a Brelot harmonic space. Let \(q_y(x)\) be the kernel on \(\Omega\) which behaves like the Newtonian kernel (if \(\Omega\) has potentials \(>0\)) or the logarithmic kernel (if there are no positive potentials on \(\Omega\)). Then, given any harmonic function \(u\) outside a compact set in \(\Omega\), then exists a signed measure \(\mu\) with compact support and a harmonic function \(H\) on \(\Omega\), such that \[ u(x)= \int_\Omega q_y(x)\,d\mu(y)+ H(x) \] outside a compact-set.
Mathematics Subject Classification ID: 31D05
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 31B10
zbMATH DE Number: 1901232
zbMATH Keywords: Bôcher's theorem
zbMATH Keywords: harmonic space
zbMATH Keywords: Newtonian kernel
zbMATH Keywords: logarithmic kernel
zbMATH Keywords: harmonic function
author: Suad AlHemedan
author: Moustafa K. Damlakhi
reviewed by: Victor Anandam
MaRDI profile type: Publication
full work available at URL: a:1020571803677
OpenAlex ID: W123143628
description: scientific article; zbMATH DE number 1901232