EXTERIOR REGULARIZED HARMONIC AND HARMONIC FUNCTIONS

This thesis mainly studies harmonic functions on exterior regions, having compact, Lipschitz boundaries, in our new finite energy function space, via the sequence of exterior harmonic Steklov eigenvalues and associated eigenfunctions. The new space is strictly larger than the standard Sobolev-Hilbert H-space, as it only needs square integrability of the gradients of the functions involved. Our results generalize exactly certain well-known results on Laplace's spherical harmonics in classical mathematical physics.