  We consider Backus's problem in geophysics. This consists in reconstructing a
harmonic potential outside the Earth when the intensity of the related field is
measured on the Earth's surface. Thus, the boundary condition is (severely)
nonlinear. The gravitational case is quite understood. It consists in the local
resolution near a monopole, i.e. the potential generated by a point mass. In
this paper, we consider the geomagnetic case. This consists in linearizing the
field's intensity near the so-called dipole, a harmonic function which models
the solenoidal potential of a magnet. The problem is quite difficult, because
the resolving operator related to the linearized problem is generally
unbounded. Indeed, existence results for Backus's problem in this framework are
not present in the literature. In this work, we locally solve the geomagnetic
version of Backus's problem in the axially symmetric case. In mathematical
terms, we show the existence of harmonic functions in the exterior of a sphere,
with given (boundary) field's intensity sufficiently close to that of a dipole
and which have the same axial symmetry of a dipole. We also show that unique
solutions can be selected by prescribing the average of the potential on the
equatorial circle of the sphere. We obtain those solutions as series of
spherical harmonics. The functional framework entails the use of fractional
Sobolev Hilbert spaces on the sphere, endowed with a spectral norm. A crucial
ingredient is the algebra structure of suitable subspaces.
