Continuity of the capacity in the class of capacitors with uniformly perfect plates
Let \(\overline\mathbb{R}^n=\mathbb{R}^n\cup\{\infty\}\), and let comp \(\overline \mathbb{R}^n\) denote the space of all non-empty compact subsets of \(\overline \mathbb{R}^n\) equipped with the Hausdorff distance induced by the spherical distance in \(\overline \mathbb{R}^n\) [see \textit{K. Kuratowski}, `Topology, Volume 1', New York (1966; Zbl 0158.40802), p. 223]. A sequence of capacitors \(\{(E_k,F_k)\}\) is said to converge to the capacitor \((E,F)\) if \(E_k\to E\) and \(F_k\to F\) in comp \(\overline\mathbb{R}^n\) as \(k\to\infty\). The author proves that \[ \text{cap} (E_k,F_k)\to \text{cap}(E,F) \text{ as }k\to \infty, \] for capacitors whose plates belong to the class of uniformly perfect sets defined by \textit{C. Pommerenke} [Arch. Math. 32, 192-199 (1979; Zbl 0393.30005] in the plane.
scientific article; zbMATH DE number 1557252