Backus problem in geophysics: a resolution near the dipole in fractional Sobolev spaces

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.