Generalized condensers and the asymptotics of their capacities under degeneration of some plates
Let us investigate a Jordan domain \(B\) in the complex plane \(\mathbb C\), and denote by \(\overline B\) (resp. \(\partial B\)) the closure of \(B\) (resp. topological boundary of \(B\)) in \(\overline{\mathbb C}\) (\(\mathbb C\cup\infty\), the Riemann sphere). Topology in \(\overline B\) is always understood to be that induced by \(\overline{\mathbb C}\) on this set. A \textit{condenser} \(C\) (\(\subset \overline B\)) is a triple \((B,\xi,\Delta)\), in which, \(E:=\{E_k\}_{k=1}^n\) is a collection of pairwise disjoint sets \(E_k\) (\(k=1,\dots,n\)), closed in \(\overline B\), \(\Delta=(\delta_k)_{k=1}^n\) is a set of \(n\) given real numbers. We denote it by \(C:= (B,\xi,\Delta)\) and call \(\xi\) the set of plates of the condenser \(C\). The \textit{capacity} of the condenser \(C\), denoted by \(\text{Cap}(C)\), is defined as the greatest lower bound (w.r.t. \(v\)) of the following Dirichlet integrals, \[ I (v; B) = \iint_B |\nabla v|^2 \,dx\,dy ,\quad v\in\mathcal F, \] where \(\mathcal F\) denotes the class of functions continuous in \(\overline B\), locally Lipschitz in \(B\) except of a finite number of points and furthermore constant and equal to \(\delta_k\) on \(E_k\) (\(k = 1,\dots, n\)). \(\mathcal F\) is called the class of \textit{admissible functions} in \(B\). In this paper the author continues his study (with others) on the asymptotic behaviour of the condenser's capacity in case some of its plates (including plates on \(\partial B\)) degenerate to a finite number of points. Under suitable conditions the author proves, grosso modo from Theorem 1 to Theorem 6 of Sect. 1, that \(\text{Cap}(\cdot)\) is indeed a conformal invariant Choquet's capacity. Sect. 2 is mainly reserved to the investigation of Robin's function and its analog with a pole at the boundary, and also to the notion of Robin's capacity. The asymptotic formula (formula 7) for \(\text{Cap}(\cdot)\) is given in the last Sect. 3 of the paper. The author notes that: 1) The formula 7 implies an asymptotic formula for the set-function ``1/Cap'', which gives rise to the notion of condenser's reduced module, studied by Kuz'mina, Emel'yanov, Solynin and others. 2) Both methods (the one given by the author and that mentioned above) are applied to study extremal problems for analytic function of one variable and related problems in the field of logarithmic potential theory.
scientific article; zbMATH DE number 5312791