A non-standard boundary value problem related to geomagnetism

Abstract

Consider the following boundary value problem in the exterior space S ^ d − 1 = { x ∈ d : | x | > 1 } {\hat S^{d - 1}} = \left \{ {x \in {^d}:\left | x \right | > 1} \right \} of a sphere S d − 1 {S^{d - 1}} in two and three dimensions ( d = 2 , 3 ) \left ( d = 2, 3 \right ) : Given a vector field D : S d − 1 → d D:{S^{d - 1}} \to {^d} we ask for all harmonic vector fields B : S ^ d − 1 → d B:{\hat S^{d - 1}} \to {^d} which decay at least as fast as a dipole field at infinity and are parallel to D on S d − 1 {S^{d - 1}} i.e. there is f : S d − 1 → f:{S^{d - 1}} \to such that B = f D B = fD . For d = 3 d = 3 , this problem is related to the problem of reconstructing the geomagnetic field outside the earth from directional data measured on the earth’s surface. The question for uniqueness or non-uniqueness is of particular interest here.