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 R N , 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 H 1 ( 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 .