Representations of Solutions of Laplacian Boundary Value Problems on Exterior Regions

Abstract

This paper treats the well-posedness and representation of solutions of Poisson's equation on exterior regions UR 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 stan- dard 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 har- monic functions and also of certain boundary trace spaces. Representations of solu- tions 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.