Topological vector spaces of harmonic functions and the trace operator

Many problems in physical geodesy can be formulated in terms of boundary-value problems (BVPs) for the gravitational potential; many of them can be ultimately formulated as a Dirichlet problem. For this reason, there is a flourishing literature of geodetic contributions to potential theory. In this paper, the authors pick up some classical arguments from the mathematical analysis of BVPs and show, by using only Hilbert spaces of harmonic functions, how they can be systematically cast into a functional scheme clarifying the role of duality when dealing with the harmonic subspaces of classical Sobolev spaces, of any real order. The analysis is here restricted to the case of functions harmonic in spherical domains to make the results transparent and more readable by geodesists. A further step is then taken showing how to generalize the Dirichlet problem for the space of all the functions that are harmonic outside a sphere, which exploits the more general theory of Fréchet topological spaces. Basically, the result is that any functions harmonic in the exterior of a sphere can be uniquely identified by a suitably defined trace on the sphere. The paper concludes with comments and discussion of future work.