On the universal solvability of classical boundary-value problems of potential theory: a contribution from geodesy

The theory of elliptic boundary-value problems in Hilbert spaces has been extensively illustrated some decades ago, e.g., by J.L. Lions and E.Magenes for very general differential operators, with coefficients, right-hand sides of the equations and boundary conditions belonging to irregular function or distribution spaces; consequently solutions too are defined in some generalized sense and belong in general to distribution spaces. In Laplace equation, on the contrary, with constant coefficients and zero right-hand side, maximal regularity properties are met inside the domain of harmonicity. It is therefore interesting, alng with the usual regularization procedures of the general theory, to develop an autonomous scheme that, making use of these regularity properties, allows to define a general topological structure for the space of the solutions of the Laplace equation in an open set. It is shown that they can therefore be classified according to the regularity of their boundary conditions, formulating suitable trace theorems. New Hilbert spaces of harmonic functions are then defined, which are different and in a sense complementary to the spaces described by Lions and Magenes. The results can be very simply proved and illustrated in the case of a spherical boundary, for which it is possible to use explicit spherical harmonic representations, and can be generalized to the case of an arbitrary regular boundary. As a matter of fact one can see that, endowing with a general topological structure the space of all the harmonic functions in an open simply connected smooth set, all these results are quite naturally generalized in a comprehensive theory of Dirichlet and Neumann problems for the Laplace operator.