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On computation of capacities and conformal invariants

Mohamed M S Nasser, Matti Vuorinen

Abstract

We give a survey of computation of the conformal capacity of planar condensers, generalized capacity, and logarithmic capacity with emphasis on our recent work 2020-2025. We also discuss some applications of our method based on the boundary integral equation with the generalized Neumann kernel to the computation of several other conformal invariants: harmonic measure, modulus of a quadrilateral, reduced modulus, hyperbolic capacity, and elliptic capacity. Here the solution of mixed Dirichlet-Neumann boundary value problem for the Laplace equation has a key role. At the end of the paper we give a topicwise structured list to our extensive bibliography on constructive complex analysis and potential theory.

On computation of capacities and conformal invariants

Abstract

We give a survey of computation of the conformal capacity of planar condensers, generalized capacity, and logarithmic capacity with emphasis on our recent work 2020-2025. We also discuss some applications of our method based on the boundary integral equation with the generalized Neumann kernel to the computation of several other conformal invariants: harmonic measure, modulus of a quadrilateral, reduced modulus, hyperbolic capacity, and elliptic capacity. Here the solution of mixed Dirichlet-Neumann boundary value problem for the Laplace equation has a key role. At the end of the paper we give a topicwise structured list to our extensive bibliography on constructive complex analysis and potential theory.

Paper Structure

This paper contains 51 sections, 2 theorems, 151 equations, 16 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Hyperbolic geometry
  4. Special functions.
  5. Grötzsch condenser
  6. Canonical domains
  7. FEM methods
  8. Extremal problems of Grötzsch and Teichmüller
  9. The boundary integral equation with the generalized Neumann kernel
  10. The integral equation
  11. Numerical solution of the integral equation
  12. Domains with corners
  13. Computation of the capacity of generalized condensers
  14. Capacity of generalized condensers
  15. The numerical method
  16. ...and 36 more sections

Key Result

Theorem 3.1

For a given function $\gamma\in H$, there exits a unique function $\rho\in H$ and a unique piecewise constant function with real constants $\nu_0,\nu_1,\ldots,\nu_m$, such that the formula defines the boundary values of an analytic function $f$ in $G$ with $f(\infty)=0$ for unbounded $G$. The function $\rho$ is the unique solution of the integral equation and the piecewise constant function $\n

Figures (16)

  • Figure 1: The domains of the condensers considered in Examples \ref{['ex:sq']}, \ref{['ex:lens']}, and \ref{['ex:dpg']}.
  • Figure 2: The relative error in the computed values of the capacities of the condensers in Examples \ref{['ex:ring']} (left) and \ref{['ex:sq']} (right) where $n$ is the number of mesh points on each boundary component of $G$.
  • Figure 3: The computed values of the capacity of the condenser $(\Omega,E)$ in Examples \ref{['ex:dpg']} and the capacities of $(\Omega,\overline{B^2(0,r_k)})$, $k=1,2$.
  • Figure 4: Left: The computed values of the capacity of the condenser in Examples \ref{['ex:lens']}. Right: absolute values of the differences between the computed values of the capacity and the estimate \ref{['eq:len-est']}.
  • Figure 5: Left: The domain of the condenser in Examples \ref{['ex:7d']}. Right: The computed values of the constants $a_1,\ldots,a_7$ for $0.01\le r\le 0.17$.
  • ...and 11 more figures

Theorems & Definitions (27)

  • Theorem 3.1: nv1
  • Remark 3.1
  • Example 4.1: Circular ring
  • Example 4.2: Square in square: bsvps10p2
  • Example 4.3: Disk with a polygonal hole
  • Example 4.4: Lens shaped plate: hnv0mc
  • Example 4.5: A disk with $7$ circular holes
  • Example 5.1: Circular domain
  • Example 6.1: Hyperbolic capacity of an ellipse
  • Remark 6.1
  • ...and 17 more