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The Eigenvalue Problem for the Laplacian via Conformal Mapping and the Gohberg--Sigal Theory

Marius Beceanu, Jiho Hong, Hyun-Kyoung Kwon, Mikyoung Lim

TL;DR

An asymptotic formula for the Laplace eigenvalues with respect to the perturbation of the domain is obtained and a fully computable a priori error estimate with no assumption on the domain’s convexity is derived.

Abstract

We consider the Dirichlet and Neumann eigenvalues of the Laplacian for a planar, simply connected domain. The eigenvalues admit a characterization in terms of a layer potential of the Helmholtz equation. Using the exterior conformal mapping associated with the given domain, we reformulate the layer potential as an infinite-dimensional matrix. Based on this matrix representation, we develop a finite section approach for approximating the Laplacian eigenvalues and provide a convergence analysis by applying the Gohberg--Sigal theory for operator-valued functions. Moreover, we derive an asymptotic formula for the Laplacian eigenvalues on deformed domains that results from the changes in the conformal mapping coefficients.

The Eigenvalue Problem for the Laplacian via Conformal Mapping and the Gohberg--Sigal Theory

TL;DR

An asymptotic formula for the Laplace eigenvalues with respect to the perturbation of the domain is obtained and a fully computable a priori error estimate with no assumption on the domain’s convexity is derived.

Abstract

We consider the Dirichlet and Neumann eigenvalues of the Laplacian for a planar, simply connected domain. The eigenvalues admit a characterization in terms of a layer potential of the Helmholtz equation. Using the exterior conformal mapping associated with the given domain, we reformulate the layer potential as an infinite-dimensional matrix. Based on this matrix representation, we develop a finite section approach for approximating the Laplacian eigenvalues and provide a convergence analysis by applying the Gohberg--Sigal theory for operator-valued functions. Moreover, we derive an asymptotic formula for the Laplacian eigenvalues on deformed domains that results from the changes in the conformal mapping coefficients.
Paper Structure (19 sections, 26 theorems, 183 equations)

This paper contains 19 sections, 26 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Operator Generalizations of Complex-Theoretic Results
  4. Exterior Conformal Mapping and Geometric Density Basis Functions
  5. Approximations of $\mathcal{K}_\Omega^\omega$ by Finite Matrices
  6. Finite Section Method for $\mathcal{K}_\Omega^\omega$
  7. Convergence Estimate assuming a greater regularity of $\partial\Omega$
  8. Relevant Lemmas
  9. Proof of Theorem \ref{['theorem:main:smooth']}
  10. Matrix representation of $P_n(T_n\mathcal{K}^{\omega}_{\Omega})P_n$
  11. Approximation of Laplacian Eigenvalues
  12. Application of the Gohberg--Sigal Theory to finite section operators of $\mathcal{K}_\Omega^\omega$
  13. Proofs of Theorems \ref{['theorem:main:first']} and \ref{['theorem:main:second']}
  14. Approximation of Eigenfunctions
  15. Shape derivative for Eigenvalues
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\partial\Omega$ be of class $C^{1,\alpha}$ for some $0<\alpha<1$. For a Neumann eigenvalue $\omega_0^2 \neq 0$ of $-\Delta$ on $\Omega$ of multiplicity $\mu$, the following hold: where the $\omega_{n,j}$ ($j=1,\dots,\mu$) denote the characteristic values of $-\frac{1}{2}I+T_n\mathcal{K}_{\Omega}^\omega$.

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Gohberg--Sigal Gohberg:1971:OEL
  • Theorem 2.2: Gohberg--Sigal Gohberg:1971:OEL
  • Theorem 2.3
  • Theorem 2.4: Jung:2021:SEL
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 36 more