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

## TL;DR

This paper explores the boundary behavior of harmonic solutions for the exterior boundary value problems in potential theory, linking it with ergodic inverse problems, and introduces a novel open problem related to Robin's problem.

## Contribution

It introduces a new open problem in potential theory's third boundary value problem and discusses the limitations of semigroup methods for its solution.

## Key findings

- Connections between boundary behaviors and ergodic inverse problems
- Use of Drazin-like inverse operator in ergodic theory context
- Identification of challenges in applying semigroup methods to Robin's 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.