On a method of deriving poles of harmonic functions on compact sets whose complements are John's sets (Q2734670)

Let \(K\) be a compact set in \(\mathbb{R}^3\) and \(\Omega= \mathbb{R}^3\setminus K\) be a John's set. Denote by \(B(0,r)\) the ball \(\{x\in \mathbb{R}^3: \|x\|< r\}\). Suppose \(K\subset B(0,r)\) and \(y\) is a point in \(B(0,3r)\setminus K\). For any \(\varepsilon> 0\) the author constructs function \(Q_{y,\varepsilon} (x)\) such that NEWLINE\[NEWLINE\sup_{x\in K} \bigl||x-y|^{-1}- Q_{y,\varepsilon} (x) \bigr|< \varepsilon,NEWLINE\]NEWLINE and \(Q_{y,\varepsilon} (x)\) is harmonic in \(B(0,3r)\). The author also gives an upper bound of the rate of growth of \(Q_{y,\varepsilon} (x)\) in \(B(0,3R)\). NEWLINENEWLINENEWLINEReviewer's remark: For related results see a paper by \textit{V. V. Andrievskij} [Ukr. Mat. Zh. 41, 1165-1169 (1989; Zbl 0695.31001)].