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Spectral completeness of the declared PEC waveguide and cavity classes

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $Ω \subset R^{3}$ be a bounded $C^{\infty}$ domain in the local graph sense of Bounded C^k domains and boundary charts, with outward Euclidean unit normal field $ν$ on $\partial Ω$ . Restate the class conventions of Spectral completeness of the curl form in a smooth PEC cavity: $L^{2} : = L^{2} (Ω; R^{3})$ is the real Hilbert space of square integrable vector fields with the inner product $⟨ u, v ⟩ = \int_{Ω} u \cdot v d x$ induced by the Euclidean inner product of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , and $H (curl) = {u \in L^{2} : curl u \in L^{2}}, H (div) = {u \in L^{2} : div u \in L^{2}}$ are the graph spaces with norms $∥ u ∥_{H (curl)}^{2} = ∥ u ∥_{2}^{2} + ∥ curl u ∥_{2}^{2}$ and $∥ u ∥_{H (div)}^{2} = ∥ u ∥_{2}^{2} + ∥ div u ∥_{2}^{2}$ , the curl and divergence being taken in the distributional sense; $X$ is the graph-norm closure in $H (curl) \cap H (div)$ of the fields that are smooth on a neighbourhood of $\overline{Ω}$ and satisfy $ν \times u = 0$ pointwise on $\partial Ω$ ; and $V = {u \in X : div u = 0}$ , $H = {\overline{V}}^{L^{2}}$ . This is the explicit weak perfect-electric-conductor class of that item, not an identification with every maximal tangential-trace realisation.

Then:

(a) Smooth PEC cavity. The curl form $(u, v) \mapsto \int_{Ω} curl u \cdot curl v$ on $V$ has a complete orthonormal $L^{2}$ eigenbasis ${u_{j}}$ with nonnegative eigenvalues $λ_{j}$ of finite multiplicity, with $λ_{j} \to \infty$ whenever $H$ is infinite dimensional and with the finite-dimensional zero eigenspace retained. Each eigenfield $u_{j} \in V$ satisfies $\int_{Ω} curl u_{j} \cdot curl v = λ_{j} \int_{Ω} u_{j} \cdot v$ for every $v \in X$ ; the eigen-equation $curl curl u_{j} = λ_{j} u_{j}$ holds distributionally together with the PEC trace; and $u_{j}$ has a $C^{\infty}$ representative on $\overline{Ω}$ for which $div u_{j} = 0$ , $ν \times u_{j} = 0$ and $\partial_{ν} u_{j, n} + (div ν) u_{j, n} = 0$ hold classically on $\partial Ω$ , where $u_{j, n} = u_{j} \cdot ν$ and $ν$ is extended constant along collar normal lines. For every explicitly chosen positive mathematical speed $v_{w a v e} > 0$ and every pair of data $q_{0} \in V$ , $q_{1} \in H$ , the associated second-order wave has coefficients solving $a_{j}^{''} + v_{w a v e}^{2} λ_{j} a_{j} = 0$ with $a_{j} (0) = ⟨ q_{0}, u_{j} ⟩$ and $a_{j}^{'} (0) = ⟨ q_{1}, u_{j} ⟩$ , in the modal class of that item: its displacement lies in $C (R; V)$ and its velocity in $C (R; H)$ , finite modal sums converge in $C (K; V)$ and $C (K; H)$ on every compact time interval $K$ , and the finitely many $λ_{j} = 0$ modes are $a_{j} (t) = a_{j} (0) + t a_{j}^{'} (0)$ . The speed is a positive abstract mathematical parameter and has no physical dependency. The scalar auxiliary Dirichlet resolvent used inside the recorded cavity argument has, for every $q \in L^{2} (Ω)$ , a unique solution $φ \in H_{0}^{1} (Ω) \cap H^{2} (Ω)$ of $(1 - Δ) φ = q$ , with smooth zero-boundary approximants converging in $H^{2}$ and whose gradients converge in the PEC graph norm, by The L-two Dirichlet resolvent and smooth zero-boundary test density. For smooth $q$ this solution is smooth on the closure by the same smooth-boundary bootstrap; no classical regularity for a general $L^{2}$ datum is asserted. Smooth unshifted scalar boundary problems have the classical upgrade of Weak and classical solvability of the elliptic boundary problems with Dirichlet, Neumann and transmission data with $ε = I$ .

(b) Rectangular reflection-parity class. Let $Q = \prod_{j = 1}^{3} (0, L_{j})$ with $L_{j} > 0$ and let $ε, μ > 0$ be constants. In the reflection boundary class of Rectangular PEC reflection class: Fourier completeness, weak solutions and energy conservation (real $L^{2}$ fields on the torus $\prod_{j} R / (2 L_{j} Z)$ whose components carry the parities of that item and whose distributional divergences vanish), the finite separated Fourier modes are dense at every time, and every initial datum of the class has a unique $C (R; L^{2})$ weak solution of $\partial_{t} E = (ε μ)^{- 1} curl B$ , $\partial_{t} B = - curl E$ in the class, along which $\frac{1}{2} \int_{Q} (ε ∣ E ∣^{2} + ∣ B ∣^{2} / μ) d x$ is constant.

No completeness is asserted in lossy, open-ended or non-smooth classes, and the constant-Ohmic and open-longitudinal branches are homed separately as physical items on the guide page; they are not mathematical premises or dependencies of this statement.

Facts & Assumptions

Given: The Axiom of Choice; a bounded $C^{\infty}$ domain $Ω \subset R^{3}$ with outward unit normal $ν$ ; the graph spaces $H (curl)$ , $H (div)$ and the class $X$ , subspace $V$ and space $H$ of the Statement; a positive speed $v_{w a v e}$ and data $q_{0} \in V$ , $q_{1} \in H$ ; positive constants $L_{j}$ , $ε$ , $μ$ for part (b).

[L1]

Smooth PEC cavity spectral closure (Spectral completeness of the curl form in a smooth PEC cavity): for a bounded $C^{\infty}$ domain $Ω$ , with $X$ the graph-norm closure in $H (curl) \cap H (div)$ of the smooth-on-closure fields with $n \times u = 0$ , $V = {u \in X : div u = 0}$ and $H = {\overline{V}}^{L^{2}}$ , the curl form on $V$ has a complete orthonormal $L^{2}$ eigenbasis with nonnegative eigenvalues $λ_{j}$ of finite multiplicity tending to infinity whenever $H$ is infinite dimensional, the zero eigenspace is finite dimensional and retained; each eigenfield $u_{j} \in V$ satisfies the full graph-class variational identity $\int_{Ω} curl u_{j} \cdot curl v = λ_{j} \int_{Ω} u_{j} \cdot v$ for every $v \in X$ and the distributional eigen-equation $curl curl u_{j} = λ_{j} u_{j}$ with the PEC trace; and for every explicitly chosen mathematical speed $v_{w a v e} > 0$ and all $q_{0} \in V$ , $q_{1} \in H$ the associated second-order wave has coefficients $a_{j}^{''} + v_{w a v e}^{2} λ_{j} a_{j} = 0$ , displacement in $C (R; V)$ , velocity in $C (R; H)$ , finite modal sums converging in $C (K; V)$ and $C (K; H)$ on every compact time interval $K$ , and affine $λ = 0$ modes $a_{j} (t) = a_{j} (0) + t a_{j}^{'} (0)$ . The speed is an abstract positive mathematical parameter.

[L2]

Eigenfield boundary regularity (Smooth boundary regularity of PEC cavity eigenfields in the declared graph completion): if $u \in X$ with $X$ as in [L1], $div u = 0$ weakly and $λ \geq 0$ satisfies $\int_{Ω} curl u \cdot curl v = λ \int_{Ω} u \cdot v$ for every $v \in X$ , then $u \in H^{m} (Ω; R^{3})$ for every integer $m \geq 1$ and $u$ has a $C^{\infty}$ representative on the closure for which $curl curl u = λ u$ , $div u = 0$ , $ν \times u = 0$ and $\partial_{ν} u_{n} + (div ν) u_{n} = 0$ hold classically, with $u_{n} = u \cdot ν$ and the normal extended constant along collar normal lines; zero eigenvalues and disconnected domains are included.

[L3]

Scalar elliptic boundary existence (Weak and classical solvability of the elliptic boundary problems with Dirichlet, Neumann and transmission data): for a bounded $C^{\infty}$ domain, uniformly positive smooth coefficient $ε$ and smooth right-hand side $ρ$ on the closure, the weak Dirichlet problem with smooth boundary data and the weak compatible Neumann problem each have exactly one solution modulo the stated normalisation, which is $C^{\infty}$ on the closure and satisfies the classical equation and boundary values; the existence content is that of the weak solvability lemmas and edge, junction and non-smooth regimes are excluded.

[L4]

Rectangular PEC energy completeness (Rectangular PEC reflection class: Fourier completeness, weak solutions and energy conservation): for $Q = \prod_{j} (0, L_{j})$ with $L_{j} > 0$ and constants $ε, μ > 0$ , in the reflection boundary class with distributional $div E = div B = 0$ the finite separated Fourier modes are dense and the class is $L^{2}$ -complete, every initial datum has a unique $C (R; L^{2})$ weak solution of $\partial_{t} E = (ε μ)^{- 1} curl B$ , $\partial_{t} B = - curl E$ in the class, and the energy $\frac{1}{2} \int_{Q} (ε ∣ E ∣^{2} + ∣ B ∣^{2} / μ) d x$ is constant along it; the item assumes Countable Choice and supplies neither trace regularity for arbitrary $H (curl)$ fields nor completeness in lossy, open or non-rectangular guides.

[L5]

Choice principles: the Axiom of Choice states that every family of nonempty sets has a choice function (The Axiom of Choice); it implies the Axiom of Dependent Choice and the Axiom of Countable Choice ${A C}_{ω}$ (AC implies DC implies countable choice), the latter meaning that every sequence of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[L6]

Conventions: a map is $C^{\infty}$ on the closure when it is the restriction of a smooth map on a neighbourhood of the closure ( $C^{k}$ Euclidean maps and diffeomorphisms); the Euclidean inner product is that of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ; and the classical product, sum and chain rules used to read the differentiated identities of [L2] are those of Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ and The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ .

[L7]

Exact scalar auxiliary resolvent and density (The L-two Dirichlet resolvent and smooth zero-boundary test density): for every $q \in L^{2} (Ω)$ there is a unique $φ \in H_{0}^{1} \cap H^{2}$ with $(1 - Δ) φ = q$ , $∥ φ ∥_{H^{2}} \leq C_{Ω} ∥ q ∥_{2}$ , and smooth zero-boundary approximants converge in $H^{2}$ , with their gradients converging in the PEC curl/div graph norm. No classical smoothness is asserted for a general $q$ .

Choice accounting. The Axiom of Choice is assumed in the Statement. It is used by [L1]–[L3], whose statements declare it, and it supplies through [L5] the Countable Choice assumed by the rectangular supplier [L4]. No further choice principle is used, and the wrapper introduces no selection.

Proof

technique · direct

Given: The Axiom of Choice, the bounded $C^{\infty}$ domain $Ω$ , the classes $X$ , $V$ , $H$ and the data of the Statement.

1.1L1L5

The definitions agree and the cavity conclusions apply. The class $X$ , the subspace $V$ and the space $H$ of the Statement are exactly those of [L1] for the same domain $Ω$ : the same graph-norm closure of the same smooth fields on the closure with $ν \times u = 0$ , the same divergence-free subspace and the same closed $L^{2}$ span, and the Axiom of Choice assumed in the Statement is the choice assumption of [L1], whose meaning is fixed by [L5]. Hence all conclusions of [L1] hold in the present setting: the curl form on $V$ has a complete orthonormal $L^{2}$ eigenbasis ${u_{j}}$ with nonnegative eigenvalues $λ_{j}$ of finite multiplicity and $λ_{j} \to \infty$ whenever $H$ is infinite dimensional, the zero eigenspace is finite dimensional and retained, each eigenfield satisfies the full $X$ variational identity and the distributional eigen-equation with the PEC trace, and the modal wave expansion with the chosen positive speed $v_{w a v e}$ has the stated initial coefficients, the stated continuities on compact intervals and affine zero modes.

1.2L4L5

Rectangular class. [L4] states exactly the assertions of part (b) for the reflection boundary class with constants $ε, μ > 0$ : density of the finite separated Fourier modes at every time, unique $C (R; L^{2})$ weak solutions of the two equations in the class for every initial datum, and constancy of $\frac{1}{2} \int_{Q} (ε ∣ E ∣^{2} + ∣ B ∣^{2} / μ) d x$ . Its Countable Choice hypothesis is available in the present setting: the Statement assumes the Axiom of Choice, and the implication to Countable Choice is the choice supplier [L5].

2.1L2L6step 1.1

Smooth eigenfields. Fix $j$ . By step 1.1, $u_{j} \in X$ , $div u_{j} = 0$ weakly and the full $X$ variational identity holds for $u_{j}$ with $λ_{j} \geq 0$ ; hence the hypotheses of [L2] are satisfied and $u_{j} \in H^{m} (Ω; R^{3})$ for every integer $m \geq 1$ with a $C^{\infty}$ representative on $\overline{Ω}$ , for which the classical relations $curl curl u_{j} = λ_{j} u_{j}$ , $div u_{j} = 0$ , $ν \times u_{j} = 0$ and $\partial_{ν} u_{j, n} + (div ν) u_{j, n} = 0$ hold, with $u_{j, n} = u_{j} \cdot ν$ and $ν$ extended constant along collar normal lines; the Euclidean pairing and the reading of the differentiated identities are those of [L6]. In particular the zero eigenvalues $λ_{j} = 0$ are included and no eigenfield is discarded.

2.2L2L3L7step 1.1

Exact scalar auxiliary boundary problem. The actual divergence-preservation argument in [L1] sets $q = div K_{0} f \in L^{2}$ and uses $(1 - Δ) φ = q$ with $φ \in H_{0}^{1} \cap H^{2}$ . Its existence and the admissible smooth zero-boundary approximation of $\nabla φ$ are exactly [L7], so the full graph-class identity can be passed to this rough-data test without assuming that $q$ is smooth. If $q$ is smooth, the boundary quotient argument recorded in [L7] and the smooth-domain bootstrap behind [L3] apply with the additional zeroth-order term $+ φ$ : tangential differentiation creates only lower-order derivatives of $φ$ , which are already controlled at each induction stage, and normal recovery uses the positive coefficient of $\partial_{n n} φ$ as before. Starting from the $H^{2}$ solution, induction gives every $H^{m}$ , hence the smooth representative on the closure; the unchanged lower-order term requires no new boundary data. Independently, [L3] supplies the classical upgrade for smooth unshifted Poisson Dirichlet or compatible Neumann problems with $ε = I$ . The vector eigenfield upgrade is still supplied by [L2], and no smoothness of the actual rough auxiliary datum is used in spectral closure.

3.1givenL5step 1.1step 1.2step 2.1step 2.2algebra∎

Assembly, exclusions and choice accounting. Part (a) is steps 1.1, 2.1 and 2.2: step 1.1 transfers the spectral completeness, multiplicity, growth, zero-mode retention, full- $X$ identity, distributional eigen-equation and modal wave expansion from [L1]; step 2.1 upgrades every eigenfield to a $C^{\infty}$ field on the closure satisfying the classical PEC relations, including the zero eigenvalues; and step 2.2 records the exact rough-data $H^{2}$ resolvent and graph-density prerequisite [L7], its smooth-data bootstrap, and the separate smooth unshifted scalar upgrade [L3]. Part (b) is step 1.2, which is [L4] verbatim in the present notation. Nothing outside the declared classes is asserted: no lossy, open-ended or non-smooth spectral completeness, no identification with every maximal tangential-trace realisation, and no physical reading of the abstract speed $v_{w a v e}$ ; the constant-Ohmic and open-longitudinal branches are homed on the guide page as separate physical items and are not used here. The Axiom of Choice is used only through [L1]–[L3] and, through its Countable Choice consequence [L5], through [L4]; no other choice principle and no choice of data beyond the explicit open set ${v_{w a v e} > 0}$ occurs. This proves all assertions.

Spectral completeness of the declared PEC waveguide and cavity classes

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources