Curl curl versus Dirichlet Laplacian eigenvalues

TL;DR

This work establishes a sharp comparison between curl curl eigenvalues from the Maxwell problem and Dirichlet Laplacian eigenvalues on bounded Lipschitz domains. Using a variational framework built on the Costabel–Dauge bilinear form, the authors prove that $\alpha_{2k+1} \leq \lambda_k$ for all $k$, with strict inequality under polyhedral boundary conditions; also, $\alpha_3 < \lambda_1$ for Lipschitz domains. The approach relies on a spectral decomposition of a combined operator $B$ whose eigenvalues comprise both $-\Delta_{\rm D}$ and curl curl spectra, and on constructing a $3k$-dimensional trial subspace from the Dirichlet eigenfunctions to derive a counting argument. The results advance Maxwell eigenvalue theory by linking curl curl spectra to the well-studied Dirichlet Laplacian, with potential implications for related inequalities involving Neumann Laplacians.

Abstract

We provide an upper estimate for the eigenvalues of the curl curl operator on a bounded, three-dimensional Euclidean domain in terms of eigenvalues of the Dirichlet Laplacian. The result complements recent inequalities between curl curl and Neumann Laplacian eigenvalues. The curl curl eigenvalues considered here correspond to the Maxwell eigenvalue problem with constant material parameters.

Theorems & Definitions (12)

• Theorem 1.1
• Proposition 2.1
• Remark 2.2
• Proposition 2.3
• proof
• Corollary 2.4
• Corollary 2.5
• proof
• Theorem 3.1
• proof