Classification of harmonic functions in the exterior of the unit ball (Q869795)

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 \(\Omega_{\zeta}.\)