The Third Boundary Value Problem of Potential Theory for the Exterior Ball and the Approximation behaviour of the solution; a Novel Open Problem

Abstract

The paper is concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet and Neumann problems of harmonic analysis for the three-dimensional unit ball with the corresponding behaviour of the associated ergodic inverse problems for the punched unlimited space. The basis is the theory of semigroups of linear operators mapping a Banach space $X$ into itself. The rates of approximation play a basic role. 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 U. Westphal (1970/71) and extended to a generalized setting with J. J. Koliha (2009). The novel open problem concerns the third or Robin's problem of potential theory, the solution of which is not a semigroup of operators. Hence, the semigroup methods applied to Dirichlet's or Neumann's problem cannot be applied. The authors give several hints how to overcome these difficulties.