Representations of Solutions of Laplacian Boundary Value Problems on Exterior Regions

This paper treats the well-posedness and representation of solutions of Poisson’s equation on exterior regions $U\subsetneq{\mathbb{R}}^{N}$ with N≥3. Solutions are sought in a space E ¹(U) of finite energy functions that decay at infinity. This space contains H ¹(U) and existence-uniqueness theorems are proved for the Dirichlet, Robin and Neumann problems using variational methods with natural conditions on the data. A decomposition result is used to reduce the problem to the evaluation of a standard potential and the solution of a harmonic boundary value problem. The exterior Steklov eigenproblems for the Laplacian on U are described. The exterior Steklov eigenfunctions are proved to generate an orthogonal basis for the subspace of harmonic functions and also of certain boundary trace spaces. Representations of solutions of the harmonic boundary value problem in terms of these bases are found, and estimates for the solutions are derived. When U is the region exterior to a 3-d ball, these Steklov representations reduce to the classical multi-pole expansions familiar in physics and engineering analysis.