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

TL;DR

This paper investigates the eigenvalue problem for the vector Laplacian in Maxwell's equations and demonstrates that the Faber-Krahn inequality, which holds for scalar Laplacians, does not apply in this vector setting.

Contribution

It provides the first proof that the Faber-Krahn inequality fails for the vector Laplacian related to Maxwell's equations.

Findings

Faber-Krahn inequality does not hold for the vector Laplacian

Counterexamples to the inequality in the context of Maxwell's equations

Highlights differences between scalar and vector Laplacian eigenvalue problems

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.