Elliptic, hyperbolic, and condenser capacity: geometric estimates for elliptic capacity
In 1947 Tsuji defined hyperbolic and elliptic versions of transfinite diameter of a closed set. An alternate more substantial definition of elliptic capacity was given by \textit{R. Kühnau} [J. Reine Angew. Math. 234, 216--220 (1969; Zbl 0185.32802), Geometrie der konformen Abbildung auf der hyperbolischen und der elliptischen Ebene (1974; Zbl 0278.30022)]. Starting with the problem to ask for the minimum of the capacity of a subset of the boundary set of a Jordan domain \textit{P. L. Duren} and \textit{M. M. Schiffer} [Pac. J. Math. 148, No. 2, 251--273 (1991; Zbl 0682.30015)] developed a concept of Robin capacity and later \textit{P. Duren} and \textit{J. Pfaltzgraff} [J. Math. Anal. Appl. 179, No. 1, 110--119 (1993; Zbl 0797.31009)] gave a definition of hyperbolic Robin capacity. A theory of elliptic Robin capacity has been developed by \textit{P. Duren} and \textit{R. Kühnau} [J. Analyse 89, 317--335 (2003; Zbl 1026.30023)]. This paper here continues the paper by Duren and Kühnau. It was proven there that \[ d_e(E) = exp\{-\pi\lambda_\Omega(C,C^*), \} \] where \(d_e\) is the elliptic capacity of \(E\) and \( \lambda_\Omega(C,C^*) \) is the extremal distance between \(C\) and \(C^*\). The author proves this identity for a more general situation and it is also proven that for a closed connected set \(E\) of the unit disk \[ d_e(E) \leq d_h(E) \] which was proven by Duren and Kühnau in a smoother case. The proofs are based on extremal length method and the theory of condensers.
scientific article; zbMATH DE number 5010203