Condenser capacity and hyperbolic perimeter

Abstract

We apply domain functionals to study the conformal capacity of condensers (G, E) where G is a simply connected domain in the complex plane and E is a compact subset of G. Due to conformal invariance, our main tools are the hyperbolic geometry and functionals such as the hyperbolic perimeter of E. Novel computational algorithms based on implementations of the fast multipole method are combined with analytic techniques. Computational experiments are used throughout to, for instance, demonstrate sharpness of established inequalities. In the case of model problems with known analytic solutions, very high precision of computation is observed.

Conformal capacity of hedgehogs

“…Indeed, very recently it was shown in [29] that isoperimetric inequalities in hyperbolic metric yield simple upper and lower bounds for the capacity in the case when E is a finite union of hyperbolic disks. In the subsequent work [28,30,31] these ideas were developed further, and it was also pointed out that similar ideas were also applied by F.W. Gehring [18] and R. Kühnau [25] fifty years earlier.…”

“…This project originated with the following question raised by the third-listed author of this paper. This question arose in the course of recent work [28]- [31] and it was experimentally studied in [23].…”