SYMBOLIC OPERATOR SOLUTIONS OF LAPLACE'S AND STOKES' EQUATIONS PART I LAPLACE'S EQUATION

Symbolic operator techniques are employed to derive a general solution of Laplace's equation in the infinite space external to a sphere. This is done for the case where the function vanishes on the sphere surface and arbitrary continuous asymptotic boundary data are imposed at infinity. such data being prescribed in the form of a solution of Laplace's equation that is analytic at the origin. In contrast with other standard methods for solving Laplace's equation, e. g., Green's functions, eigenfunction expansions, etc., the novelty of the proposed method lies in the fact that the solution can be expressed in a completely explicit form, directly in terms of (radial derivatives of) the given "undisturbed" field at infinity. A reciprocal theorem is derived and used to demonstrate that certain integral properties of the field can be obtained directly from the prescribed data at infinity, without recourse to a detailed solution of the relevant boundary-value problem. This global symbolic operator technique is illustrated for ellipsoidal as well as spherical particles. The elementary scalar harmonic analysis of the present paper serves as an entre to a companion paper (Part II), concerned with the application of similar symbolic techniques to the solution of more difficult vector biharmonic boundary-value problems, relevant to hydrodynamic Stokes flows in the infinite region external to a particle.