Spectral representations of solutions of linear elliptic equations on exterior regions

This paper describes spectral representations and approximations of solutions of second order, self-adjoint, linear elliptic boundary value problems on exterior regions U in RN, for N≥2. Inhomogeneous Dirichlet, Robin and Neumann boundary conditions are treated. Some new trace results are proved and used to provide spectral characterizations of the boundary traces of functions in H1(U). Orthonormal bases of these Hilbert spaces are derived using a theory of Steklov eigenproblems. The Steklov eigenfunctions of a regularized Laplace operator on the exterior of a ball are explicitly determined when N=2,3.