Condenser capacity and hyperbolic perimeter

TL;DR

This paper introduces new computational algorithms combining the fast multipole method and analytic techniques to study conformal capacities of condensers, emphasizing hyperbolic geometry and perimeter, with high-precision results validating inequalities.

Contribution

The paper develops novel computational methods for conformal capacity analysis using hyperbolic geometry and domain functionals, enhancing accuracy and efficiency.

Findings

Demonstrates sharpness of established inequalities.

Achieves high-precision computations for model problems.

Validates the use of hyperbolic perimeter in capacity estimates.

Abstract

We study the conformal capacity by using novel computational algorithms based on implementations of the fast multipole method, and analytic techniques. Especially, we apply domain functionals to study the capacities 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 $E$. Our computational experiments demonstrate, for instance, sharpness of established inequalities. In the case of model problems with known analytic solutions, very high precision of computation is observed.