  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.
