Classification of harmonic functions in the exterior of the unit ball
type: Item
instance of: scholarly article
title: Classification of harmonic functions in the exterior of the unit ball
zbMATH Open document ID: 1113.31002
DOI: 10.1023/B:MATN.0000015033.66562.08
author: P. A. Akhmet'ev
author: A. V. Khokhlov
published in: Mathematical Notes
publication date: 2007-03-09T00:00:00Z
review text: The authors investigate the nonlinear problem \[ \Delta f = 0, \] \[ | | \text{grad} f (x)| | = \psi (x) | _{| | x| | =1}, f (r) = o ({1}/{R}) (r \to \infty) \] in the exterior infinite spherical domain in \(\mathbb R^3.\) \textit{G. E. Backus} [Non-uniqueness of the external geomagnetic field determined by surface intensity measurements, J. Geophys. Res. 75, 6339--6341 (1970)] constructed a pair of different harmonic functions \(f_1, f_2\) solving the problem for some choice of the function \(\psi.\) \textit{A. Khokhlov, G. Hulot} and \textit{J. L. Moul} [On the Backus effect. I, Geophys. J. Int. 130, 701--703 (1997)] found a condition which was sufficient for the problem to have a unique solution. Let \(\Omega_{\zeta}\) be the space of pairs \(f_1, f_2\) of solutions of the problem (the function \(\psi\) is not fixed) such that \(f_1 - f_2 = \zeta.\) Under the assumptions that \(\zeta\) is a harmonic polynomial (of the variable \(\frac{1}{r}\)) and is independent of the spherical coordinate \(\varphi\) the authors describe all functions in \(\Omega_{\zeta}.\)
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35C10
zbMATH DE Number: 5132511
zbMATH Keywords: harmonic function
zbMATH Keywords: analytic function
zbMATH Keywords: Laplace equation
zbMATH Keywords: Poincaré-Perron theorem
zbMATH Keywords: power series
MaRDI profile type: Publication
full work available at URL: b:matn.0000015033.66562.08
OpenAlex ID: W2022322924
description: scientific article; zbMATH DE number 5132511