In this talk, we shall introduce the proper function space, i.e., the finite energy space E¹(U), to study the weak solvability of harmonic and modified harmonic equations on some unbounded region U⊆R³, having compact closed piecewise Lipschitz boundary ∂U. As a matter of fact, we have:

H¹(U) ⊂ ≠ E¹(U),

whose preciseness is provided, say, by the function r.

Via the exterior harmonic and modified harmonic Steklov eigenvalues and associated families of eigenfunctions, spaces of all the weak harmonic and modified harmonic functions on U are studied. Also, Poisson and Neumann-Robin kernels for solving boundary value problems in these spaces are defined, and their respective spectra are described explicitly.

On the other hand, the fractional Sobolev space H^1/2(∂U) can be characterized with an equivalent inner product by the exterior harmonic Steklov eigenvalues and eigenfunctions, and through this space, an isomorphism between the interior harmonic function space and the exterior one is found.