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Spectral geometry of the curl operator on smoothly bounded domains

Josef Greilhuber, Willi Kepplinger

TL;DR

The paper studies the curl operator on smoothly bounded domains in $\mathbb{R}^3$ under general Lagrangian boundary conditions, recasting the problem in a differential-form framework with a self-adjoint realization and discrete spectrum. A central contribution is a Hadamard-type variation formula for curl eigenvalues under domain perturbations, enabling a rigorous perturbation theory that yields generic simplicity of the spectrum for a comeagre set of domain deformations. The authors further develop a variational approach to domain optimization by deriving necessary conditions for local extrema of curl eigenvalue functionals, clarifying boundary normalization properties and extending prior results to general Lagrangian boundary data. Together, these results advance the spectral geometry of curl, provide a unified treatment of boundary conditions, and have potential implications for optimal domain design in plasma physics and related PDE contexts.

Abstract

We show that the spectrum of the curl operator on a generic smoothly bounded domain in three-dimensional Euclidean space consists of simple eigenvalues. The main new ingredient in our proof is a formula for the variation of curl eigenvalues under a perturbation of the domain, reminiscent of Hadamard's formula for the variation of Laplace eigenvalues under Dirichlet boundary conditions. As another application of this variational formula, we simplify the derivation of a well-known necessary condition for a domain to minimize the first curl eigenvalue functional among domains of a given volume and derive similar necessary conditions for a domain extremizing higher eigenvalue functionals.

Spectral geometry of the curl operator on smoothly bounded domains

TL;DR

The paper studies the curl operator on smoothly bounded domains in under general Lagrangian boundary conditions, recasting the problem in a differential-form framework with a self-adjoint realization and discrete spectrum. A central contribution is a Hadamard-type variation formula for curl eigenvalues under domain perturbations, enabling a rigorous perturbation theory that yields generic simplicity of the spectrum for a comeagre set of domain deformations. The authors further develop a variational approach to domain optimization by deriving necessary conditions for local extrema of curl eigenvalue functionals, clarifying boundary normalization properties and extending prior results to general Lagrangian boundary data. Together, these results advance the spectral geometry of curl, provide a unified treatment of boundary conditions, and have potential implications for optimal domain design in plasma physics and related PDE contexts.

Abstract

We show that the spectrum of the curl operator on a generic smoothly bounded domain in three-dimensional Euclidean space consists of simple eigenvalues. The main new ingredient in our proof is a formula for the variation of curl eigenvalues under a perturbation of the domain, reminiscent of Hadamard's formula for the variation of Laplace eigenvalues under Dirichlet boundary conditions. As another application of this variational formula, we simplify the derivation of a well-known necessary condition for a domain to minimize the first curl eigenvalue functional among domains of a given volume and derive similar necessary conditions for a domain extremizing higher eigenvalue functionals.

Paper Structure

This paper contains 13 sections, 10 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The setup
  3. Statement of results
  4. Organization of this paper.
  5. Acknowledgements
  6. Preliminary observations
  7. Various domains for the curl operator
  8. Self-adjoint realizations of the curl operator
  9. Reinterpreting the zero-flux boundary condition
  10. Analytic dependence of the resolvent
  11. The Hadamard rule
  12. Generic simplicity of the spectrum
  13. Local extrema of the eigenvalue functionals

Key Result

Theorem 1.1

Let $D_0 \subseteq \mathbb R^3$ be a smoothly bounded domain and let $L\subset H^1_{dR}(\partial D_0,\mathbb R)$ be a Lagrangian subspace. For a comeagre subset of diffeomorphisms $\Phi \in \mathcal{X}(D_0)$ the spectrum of the operator $\mathop{\mathrm{curl}}\nolimits_{L_\Phi}$ on $D_\Phi$ consists

Figures (1)

  • Figure 1: The curves $\alpha_j$ and $\beta_j$. Note that the surface $\partial D$ need not be connected.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.1
  • proof
  • Remark 3.2
  • ...and 13 more