Well‐posedness and approximation of solutions of linear divergence‐form elliptic problems on exterior regions

This paper describes well‐posedness, spectral representations, and approximations of solutions of uniformly elliptic, second‐order, divergence form elliptic boundary value problems on exterior regions U inR Nwhen N  ≥ 3. Inhomogeneous Dirichlet, Neumann, and Robin boundary conditions are treated. These problems are first shown to be well‐posed in the space E 1 ( U ) of finite‐energy functions on U using variational methods. Spectral representations of these solutions involving Steklov eigenfunctions and solutions subject to zero Dirichlet boundary conditions are described. Some approximation results for the A ‐harmonic components are obtained. Positivity and comparison results for these solutions are given. Copyright © 2014 John Wiley & Sons, Ltd.