A note on the failure of the Faber-Krahn inequality for the vector Laplacian

TL;DR

This work analyzes the Maxwell curl–curl eigenproblem with divergence-free constraint under volume and perimeter constraints and shows a failure of the Faber–Krahn inequality for the vector Laplacian. Using explicit cuboid domains, the authors obtain precise first-eigenvalue formulas and prove that $\inf_{|\Omega|=k}\lambda_1^\Omega=0$ and $\sup_{|\Omega|=k}\lambda_1^\Omega=+\infty$, while $\inf_{|\partial\Omega|=k}\lambda_1^\Omega=4\pi^2/k$ and $\sup_{|\partial\Omega|=k}\lambda_1^\Omega=+\infty$ (with the perimeter infimum not attained). They further construct three-dimensional dumbbell domains to drive the first eigenvalue to zero under fixed surface area, by exploiting a product-domain spectral formula $\lambda_1^\Omega=\min\{\mu_1^{\mathcal D},\mu_1^{\mathcal N}+\pi^2/h^2\}$ and a 2D Neumann eigenvalue decay on narrowing channels. The analysis connects to the Hodge Laplacian viewpoint and highlights the non-universality of the ball as a minimizer for vectorial Maxwell-type problems, motivating refined geometric constraints for lower bounds in spectral design of electromagnetic cavities.

Abstract

We consider a natural eigenvalue problem for the vector Laplacian related to stationary Maxwell's equations in a cavity and we prove that an analog of the celebrated Faber-Krahn inequality doesn't hold.

Figures (2)

• Figure 1: The base domain $\omega_h$
• Figure 2: the domain $\omega_{\delta , \eta}$

Theorems & Definitions (11)

• Theorem 1
• proof
• Theorem 2
• proof
• Remark 3
• Remark 4
• Remark 5
• Theorem 6
• proof
• Lemma 7