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Matrix representation of the Neumann-Poincaré operator for a torus

Doosung Choi

Abstract

We represent a matrix representation of the Neumann-Poincaré operator defined on the boundaries of a torus. A torus is a doubly connected domain in three dimensions. There is a well-known parametrization for the shape of the torus, the toroidal coordinate system. Based on the coordinate system, we use toroidal harmonics to get an expansion of the NP operator for the torus. Along with proper bases, the Neumann-Poincaré operator can be explicitly represented by an infinite matrix.

Matrix representation of the Neumann-Poincaré operator for a torus

Abstract

We represent a matrix representation of the Neumann-Poincaré operator defined on the boundaries of a torus. A torus is a doubly connected domain in three dimensions. There is a well-known parametrization for the shape of the torus, the toroidal coordinate system. Based on the coordinate system, we use toroidal harmonics to get an expansion of the NP operator for the torus. Along with proper bases, the Neumann-Poincaré operator can be explicitly represented by an infinite matrix.

Paper Structure

This paper contains 6 sections, 7 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Layer Potentials
  4. Toroidal coordinate system
  5. Toroidal harmonics
  6. Matrix representation of the NP operator for a torus

Key Result

Lemma 3.1

For $z = \cosh\tau$, the toroidal functions satisfy the Wronskian relation for all $m,n\in\mathbb{Z}$.

Theorems & Definitions (8)

  • Lemma 3.1: Lebedev:1965:SFT
  • Proposition 3.2
  • Lemma 3.3: Cohl:2011:GHI
  • Lemma 3.4
  • Lemma 3.5
  • Proposition 3.6
  • proof
  • Theorem 3.7: A matrix of the NP operator for tori