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The paper is concerned with the interconnection of the boundary behaviour ofthe solutions of the exterior Dirichlet and Neumann problems of harmonicanalysis for the three-dimensional unit ball with the corresponding behaviourof the associated ergodic inverse problems for the punched unlimited space. Thebasis 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 infinitesimalgenerator $A$ of a semigroup that arises naturally in ergodic theory. Thisoperator $B$ is a closed, not necessarily bounded, operator. It was introducedin a paper with U. Westphal (1970/71) and extended to a generalized settingwith J. J. Koliha (2009).  The novel open problem concerns the third or Robin's problem of potentialtheory, the solution of which is not a semigroup of operators. Hence, thesemigroup methods applied to Dirichlet's or Neumann's problem cannot beapplied. The authors give several hints how to overcome these difficulties.

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