Boundary value problems of potential theory for the exterior ball and the approximation and ergodic behaviour of the solutions

Abstract

The paper is concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet, Neumann and Robin problems of harmonic analysis for the unit ball in $R^3$ with the corresponding behaviour of the associated ergodic inverse problems for the entire space. Rates of approximation play a basic role.

The solutions themselves are evaluated by means of Fourier expansions with respect to spherical harmonics. In case of the first two problems, the basis for the investigation of the approximation and ergodic behaviour is the theory of semigroups of linear operators mapping a Banach space X into itself. The connection between the semigroup property and the major premise of Huygens’ principle is emphasized.

Another tool is a Drazin-like inverse operator $A^{\mathrm{ad}}$ for the infinitesimal generator $A$ of a semigroup that arises naturally in ergodic theory. This operator is a closed, not necessarily bounded, operator. It was introduced in a paper with U. Westphal (Butzer and Westphal, 1970/71) and extended to a generalized setting with J.J. Koliha (Butzer and Koliha, 2009).

Unlike the latter two problems, the solution of Robin’s problem does not have the semigroup property and therefore the semigroup methods applied to Dirichlet’s and Neumann’s problem do not work. The authors give several hints how to overcome these difficulties.