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Constrained maximization of conformal capacity

Harri Hakula, Mohamed M. S. Nasser, Matti Vuorinen

TL;DR

This work addresses maximizing the conformal capacity $\mathrm{cap}(\mathbb{B}^2,E)$ of a unit-disk condenser where $E$ is a union of $m$ disjoint hyperbolic disks (or segments) with fixed hyperbolic sizes, under center-position constraints. The authors formulate a nonlinear optimization problem and solve it using two independent numerical engines: a boundary integral equation (BIE) method with a generalized Neumann kernel and a high-order $hp$-FEM, both coupled to an interior-point optimizer. Across extensive experiments, a dispersion phenomenon emerges: components migrate as close to the unit circle as allowed by the constraints while keeping mutual distances large, with symmetric, sectorial configurations yielding additive capacity in the appropriate limit. The results offer computational insight into extremal capacity problems, validate methodological frameworks, and point to future extensions to other domains, segmentations, and alternative capacities.

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.

Constrained maximization of conformal capacity

TL;DR

This work addresses maximizing the conformal capacity of a unit-disk condenser where is a union of disjoint hyperbolic disks (or segments) with fixed hyperbolic sizes, under center-position constraints. The authors formulate a nonlinear optimization problem and solve it using two independent numerical engines: a boundary integral equation (BIE) method with a generalized Neumann kernel and a high-order -FEM, both coupled to an interior-point optimizer. Across extensive experiments, a dispersion phenomenon emerges: components migrate as close to the unit circle as allowed by the constraints while keeping mutual distances large, with symmetric, sectorial configurations yielding additive capacity in the appropriate limit. The results offer computational insight into extremal capacity problems, validate methodological frameworks, and point to future extensions to other domains, segmentations, and alternative capacities.

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 from the origin and in the second case it is required that the centers are on the subsegment 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.
Paper Structure (29 sections, 3 theorems, 51 equations, 28 figures, 10 tables)

This paper contains 29 sections, 3 theorems, 51 equations, 28 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Illustrative Example
  3. Organization
  4. Preliminaries
  5. Hyperbolic geometry
  6. Special functions
  7. Condenser capacity
  8. Modulus of a quadrilateral.
  9. Symmetric compartments.
  10. Methods
  11. High-Order Finite Element Method
  12. A Priori Refinement Strategies
  13. BIE method
  14. Domains bounded by smooth curves
  15. Domains bounded by slits
  16. ...and 14 more sections

Key Result

Lemma 1

hkv (1) If $0<a<b$ and $D=\overline{B}^2(0,b)\backslash B^2(0,a)$, (2) If $R>0$ then for $x \in \mathbb{B}^2$

Figures (28)

  • Figure 1: Surface plots of the potentials for the six hyperbolic disks with equal hyperbolic radii $0.2$. The centers of the disks are at the initial positions (left) and at the positions that maximize the capacity (right).
  • Figure 2: Sectorial symmetry. The capacity of each sector or compartment can be computed separately. In the case of a constellation with identical elements, each compartment can be divided into two quadrilaterals symmetric with respect to the radius bisecting the sector.
  • Figure 3: Mesh refinements. Left: FEM mesh with the five segments on the diameter indicated with thick lines. Middle: A detail of the mesh at one of the end points of the segments after one application of the replacement rule. Right: Successive levels of refinements, eight altogether, are shown in the plot (the smallest ones are not visible in the given scale).
  • Figure 4: Curved elements. Left: FEM mesh with six disks. Right: A detail showing the curved edges.
  • Figure 5: A schematic of a given bounded multiply connected domain $\Omega$ interior to the unit circle and exterior to $m$ radial slits (left) and its conformally equivalent computed domain $D$ interior to the unit circle and exterior to $m$ smooth Jordan curves (right) for $m=6$.
  • ...and 23 more figures

Theorems & Definitions (4)

  • Lemma 1
  • Theorem 1
  • Theorem 2: hno
  • Remark 1