Constrained maximization of conformal capacity

TL;DR

This paper investigates how to maximize the conformal capacity of disk and segment constellations within the unit disk under specific geometric constraints, using computational methods to analyze optimal configurations.

Contribution

It introduces a constrained optimization problem for conformal capacity involving hyperbolic disks and segments, and provides computational insights into optimal arrangements.

Findings

Disks tend to cluster near the boundary under constraints.

Disks and segments stay as far apart as possible within the constraints.

Computational methods effectively solve the Dirichlet problem for capacity calculation.

Abstract

We consider constellations of disks which are unions of disjoint hyperbolic disks in the unit disk with fixed radii and unfixed centers. We study the problem of maximizing the conformal capacity of a constellation with a fixed number of disks under constraints on the centers in two cases. In the first case the constraint is that the centers are at most at distance $R \in(0,1)$ from the origin and in the second case it is required that the centers are on the subsegment $[-R,R]$ of a diameter of the unit disk. We study also similar types of constellations with hyperbolic segments instead of the hyperbolic disks. Our computational experiments suggest that a dispersion phenomenon occurs: the disks/segments go as close to the unit circle as possible under these constraints and stay as far as possible from each other. The computation of capacity reduces to the Dirichlet problem for the Laplace equation which we solve using two methods: a fast boundary integral equation method and a high-order finite element method.