Exterior Dirichlet and Neumann Problems and the Linked Ergodic Inverse Problems in the Entire Space

Abstract

The paper is mainly concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet and Neumann problems of harmonic analysis for the unit disk in {mathbb {R}}^2 and the unit ball in {mathbb {R}}^3 with the corresponding behaviour of the associated ergodic inverse problems for the entire space. The basis is the theory of semigroups of linear operators mapping a Banach space X into itself. The classical one-parameter theory for semigroups applies in the present particular applications, actually for X= L^2_{2pi } in case of the unit disk, and X=L^2(S) in the three dimensional setting, S being the unit sphere in {mathbb {R}}^3. Another tool is a Drazin-like inverse operator B for the infinitesimal generator A of a semigroup that arises naturally in ergodic theory. This operator B is a closed, not necessarily bounded, operator. It was introduced in a paper with Butzer and Westphal (Indiana Univ Math J 20:1163–1174, 1970/1971) and extended to a generalized setting with Butzer and Koliha (J Oper Theory 62:297–326, 2009).