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The unquotiented zero eigenspace is the PEC curl kernel

Statement

Let $V$ , $K$ , and the unquotiented PEC curl-curl pencil be as in PEC Maxwell cavity eigenproblem and full-kernel gauge, with constant $μ > 0$ . For $u \in V$ , the weak eigenrelation at eigenvalue $0$ holds for every $v \in V$ if and only if $u \in K$ . Thus the zero eigenspace is exactly the full PEC curl kernel.

Facts & Assumptions

Given: the space $V = H_{0} (curl; Ω)$ , its curl kernel $K$ , the unquotiented weak eigenrelation, and $μ > 0$ .

[F1]

At eigenvalue $0$ , the unquotiented relation is $μ^{- 1} (curl u, curl v)_{L^{2}} = 0$ for all $v \in V$ , and $K = {u \in V : curl u = 0}$ . (PEC Maxwell cavity eigenproblem and full-kernel gauge)

Proof

Given: $u \in V$ and the weak eigenrelation at eigenvalue $0$ .

1.1F1givenalgebra

Set $v = u$ in [F1]. Then $μ^{- 1} ∥ curl u ∥_{L^{2}}^{2} = 0$ . Since $μ > 0$ , the squared norm vanishes and $curl u = 0$ , so $u \in K$ by [F1].

2.1F1givenalgebra∎

Conversely, if $u \in K$ , then [F1] gives $curl u = 0$ . Hence $μ^{- 1} (curl u, curl v)_{L^{2}} = 0$ for every $v \in V$ , which is precisely the unquotiented eigenrelation at $λ = 0$ . The two inclusions identify the zero eigenspace with $K$ .

Depends on

PEC Maxwell cavity eigenproblem and full-kernel gauge

Used by

A topological PEC zero mode on a spherical shell Example

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Sources

Long Chen, Finite Element Methods for Maxwell Equations, complete lecture notes (standard reference, not scraped)