Rectangular PEC reflection class: Fourier completeness, weak solutions and energy conservation

A smooth field on ${\mathbb{R}}^3$ satisfying the parity pattern of the statement restricts to $Q$ as a tangential-electric and normal-magnetic perfect-conductor field: at the wall $x_1=0$ (and $x_1=L_1$), $E_1$ is even in $x_1$ while $E_2,E_3$ are odd, so $E_2=E_3=0$ there, which is $n\times E=0$; $B_1$ is odd and $B_2,B_3$ even, so $B_1=0$ there, which is $n\cdot B=0$; the cyclic coordinates are identical. The parity conditions are closed linear conditions in $L^2(\mathbb{T};{\mathbb{R}}^3)$ because a sign change of a torus coordinate preserves $L^2$ convergence and distributional derivatives, and $\mathrm{div}E=0$ is the closed condition $\int E\cdot \mathrm{\nabla }\phi d\nu =0$ for every smooth periodic scalar test $\phi$, and likewise for $B$, so $\mathcal{C}$ is a closed real subspace of the Hilbert space $L^2(\mathbb{T};{\mathbb{R}}^3{)}^2$. The weak formulation used below is: $(E,B)\in C(\mathbb{R};\mathcal{C})$ is a weak solution when for every pair of smooth periodic vector fields $\mathrm{\Psi },\mathrm{\Theta }$ compactly supported in $t$, $\iint E\cdot {\mathrm{\partial }}_t\mathrm{\Psi }d\nu dt+(\epsilon \mu {)}^{-1}\iint B\cdot \mathrm{curl}\mathrm{\Psi }d\nu dt=0$ and $\iint B\cdot {\mathrm{\partial }}_t\mathrm{\Theta }d\nu dt-\iint E\cdot \mathrm{curl}\mathrm{\Theta }d\nu dt=0$, which is exactly the statement that the two equations hold for ${\mathrm{\partial }}_t$ and $\mathrm{curl}$ in the sense of Distributional derivative (the spatial sign convention in $\mathrm{curl}$ is the distributional one, and for smooth periodic fields the pairing identities of the definition reduce to the displayed ones).

The characters ${\mathrm{\Phi }}_n$ of [L1] are orthonormal in $L^2(\mathbb{T};\mathbb{C})$: orthogonality factors as a product of one-dimensional character pairings, and the $j$th factor is $1$ for equal coordinate indices and $0$ otherwise by the one-dimensional orthonormality of [L1]. They are complete: if $f\in L^2(\mathbb{T};\mathbb{C})$ is orthogonal to every ${\mathrm{\Phi }}_n$, fix $n_2,n_3\in \mathbb{Z}$ and set $g(x_1):=\int f(x_1,x_2,x_3)e^{-i\pi (n_2{x}_2\mathrm{/}{L}_2+n_3{x}_3\mathrm{/}{L}_3)}d{\nu }_{2}d{\nu }_3$, an $L^2$ function of $x_1$ by Cauchy-Schwarz; its one-dimensional character coefficients all vanish, since they are the ${\mathrm{\Phi }}_n$ pairings of $f$ with $n_1$ varying, so $g=0$ a.e. by the one-dimensional completeness of [L1]; applying this for all $n_2,n_3$ and iterating in the other two coordinates, with a countable union of exceptional null sets, gives $f=0$ a.e., so the closed span is all of $L^2(\mathbb{T};\mathbb{C})$. Parseval and mean-square convergence of finite partial sums follow from [L1] applied componentwise to the six real components of a pair of fields.

Group the character indices into the disjoint reflection orbits $O_p=\{n\in {\mathbb{Z}}^3:\mathrm{\mid }{n}_j\mathrm{\mid }=p_j(j=1,2,3)\}$ indexed by $p\in {\mathbb{N}}_0^3$. Let $P_p$ be the finite-dimensional real space of Fourier coefficient arrays $(v_n,w_n{)}_{n\in O_p}$ satisfying reality, the stated parities, and $k_n\cdot v_n=k_n\cdot w_n=0$, where $k_n=(\pi n_1\mathrm{/}{L}_1,\pi n_2\mathrm{/}{L}_2,\pi n_3\mathrm{/}{L}_3)$. The divergence constraints are equivalent to these equations: testing against each character gives necessity, and an orbit sum with these equations has zero divergence pointwise. Projection onto a whole orbit commutes with coordinate reflections and reality. Thus whole-orbit truncations lie in $\mathcal{C}$ and converge in $L^2$ by step 2.1, proving density. The corresponding real products are the stated even-coordinate cosines and odd-coordinate sines, with zero-index sine terms absent. Parseval gives $\epsilon \mathrm{\parallel }E{\mathrm{\parallel }}_2^2+\mathrm{\parallel }B{\mathrm{\parallel }}_2^2\mathrm{/}\mu ={\sum }_p{Q}_p$, where $Q_p={\sum }_{n\in O_p}(\epsilon \mathrm{\mid }{v}_n{\mathrm{\mid }}^2+\mathrm{\mid }{w}_n{\mathrm{\mid }}^2\mathrm{/}\mu )$. The mode equations are $v_n^{\mathrm{\prime }}=i(\epsilon \mu {)}^{-1}{k}_n\times w_n$, $w_n^{\mathrm{\prime }}=-i{k}_n\times v_n$. Curl sends the magnetic parity pattern to the electric pattern and conversely: differentiating changes the parity of exactly the differentiated coordinate; the two terms in each curl component have the required pattern. These equations preserve reality and divergence, so define a real linear map on $P_p$. For real $k_n$, put $C={\bar{v}}_n\cdot (k_n\times w_n)$. The scalar triple product gives ${\bar{w}}_n\cdot (k_n\times v_n)=-\bar{C}$, hence the energy derivative is $(2\mathrm{/}\mu )\mathrm{Re}(iC+i\bar{C})=0$. Therefore each $Q_p$ is conserved.

Enumerate $p\in {\mathbb{N}}_0^3$ by increasing $p_1+p_2+p_3$, then lexicographically within each finite level, writing $p^{(1)},p^{(2)},\dots$. Every coefficient belongs to exactly one $P_{p^{(m)}}$; let $A_m$ be the real matrix of the orbit operator of step 3.1 in the real basis obtained by Gaussian elimination on its finite linear constraints, with the coordinates in lexicographic order and the first available pivot chosen each time. This defines all bases without a countable choice of bases. Each $A_m$ is bounded, so the finite-dimensional linear system $z_m^{\mathrm{\prime }}=A_m{z}_m$ has a unique global solution through every real initial vector $z_m(0)$, by the global Lipschitz corollary of [L3]; no choice is needed for one system. For $M\ge 1$ let $(E_M,B_M)$ be the sum of the first $M$ modal solutions with the initial amplitudes inherited from the given initial datum $(E_0,B_0)\in \mathcal{C}$. Each $(E_M,B_M)$ is a finite sum of smooth modes, hence smooth and in $\mathcal{C}$, solves the two equations pointwise, and satisfies $\epsilon \mathrm{\parallel }{E}_M(t){\mathrm{\parallel }}^2+\mathrm{\parallel }{B}_M(t){\mathrm{\parallel }}^2\mathrm{/}\mu ={\sum }_{m\le M}{Q}_{p^{(m)}}(0)$ for all $t$ by the modal invariance of step 3.1. The omitted modes carry the nonnegative initial partial energies $R_M:={\sum }_{m>M}{Q}_{p^{(m)}}(0)$, which decrease to $0$ because the total initial energy ${\sum }_m{Q}_{p^{(m)}}(0)=\epsilon \mathrm{\parallel }{E}_0{\mathrm{\parallel }}^2+\mathrm{\parallel }{B}_0{\mathrm{\parallel }}^2\mathrm{/}\mu$ is finite. For $M<M^{\mathrm{\prime }}$ and every $t\in \mathbb{R}$, $\epsilon \mathrm{\parallel }(E_{M^{\mathrm{\prime }}}-E_M)(t){\mathrm{\parallel }}^2+\mathrm{\parallel }(B_{M^{\mathrm{\prime }}}-B_M)(t){\mathrm{\parallel }}^2\mathrm{/}\mu ={\sum }_{M<m\le M^{\mathrm{\prime }}}{Q}_{p^{(m)}}(t)\le R_M$, because each omitted modal energy is constant; hence $(E_M,B_M)$ is Cauchy uniformly in $t$ in $L^2(\mathbb{T};{\mathbb{R}}^3{)}^2$, and its uniform limit $(E,B)\in C(\mathbb{R};L^2)$ exists and takes values in the closed subspace $\mathcal{C}$ of step 1.1, with $\epsilon \mathrm{\parallel }E(t){\mathrm{\parallel }}^2+\mathrm{\parallel }B(t){\mathrm{\parallel }}^2\mathrm{/}\mu ={\lim }_M(\epsilon \mathrm{\parallel }{E}_M(t){\mathrm{\parallel }}^2+\mathrm{\parallel }{B}_M(t){\mathrm{\parallel }}^2\mathrm{/}\mu )={\sum }_m{Q}_{p^{(m)}}(0)$ for every $t$.

Let $(E,B)\in C(\mathbb{R};\mathcal{C})$ be any weak solution in the sense of step 1.1, and fix a mode index $m$ together with a real mode function $\phi$ in the finite-dimensional space of modes belonging to the orbit $O_{p^{(m)}}$ (a smooth periodic vector field). Testing the two weak equations with $\psi (t)\phi (x)$, where $\psi \in C_c^{\mathrm{\infty }}(\mathbb{R})$ is arbitrary, gives, in the sense of distributions on $\mathbb{R}$, the scalar equations $c_{\phi }^{\mathrm{\prime }}=(\text{the corresponding finite linear combination of the other modal coefficients})$, where $c_{\phi }(t)=⟨\phi ,(E,B)(t)⟩$ is the coefficient pairing; these are continuous functions of $t$ because $(E,B)$ is continuous into $L^2$ and the pairing is continuous. Assembling the finitely many coefficients belonging to $O_{p^{(m)}}$ into a real vector $Z_m(t)$ gives a finite linear system ${\mathrm{\partial }}_t{Z}_m=A_m{Z}_m$ with continuous right-hand side; applying the primitive construction componentwise on a compact interval $[a,b]$, every component of $A_m{Z}_m$ is continuous, hence Riemann integrable [L4], its oriented primitive $G$ is $C^1$ with $G^{\mathrm{\prime }}=g$ [L4], and the regular-distribution compatibility clause of [L4] (whose use here is covered by the standing ${\mathrm{A}\mathrm{C}}_{\omega }$) transfers this to distributional derivatives; subtracting the primitive from $Z_m$ shows that the difference has zero distributional derivative, hence is constant on the connected set $\mathbb{R}$ [L4]; continuity makes the equality pointwise. Therefore every modal coefficient is a classical $C^1$ solution of its finite linear system. Since $Z_m$ solves a linear constant-coefficient ODE, its value at every time is determined by $Z_m(0)$ by the global uniqueness part of [L3]; hence the weak solution $(E,B)$ is uniquely determined by its initial datum $(E(0),B(0))$, because the modal coefficients determine the fields by the completeness in step 3.1.

The energy of the constructed solution is constant. By step 4.1, for every $t$ one has $\epsilon \mathrm{\parallel }E(t){\mathrm{\parallel }}^2+\mathrm{\parallel }B(t){\mathrm{\parallel }}^2\mathrm{/}\mu ={\sum }_m{Q}_{p^{(m)}}(0)$, the same value at every $t$; and ${\sum }_m{Q}_{p^{(m)}}(0)=\epsilon \mathrm{\parallel }{E}_0{\mathrm{\parallel }}^2+\mathrm{\parallel }{B}_0{\mathrm{\parallel }}^2\mathrm{/}\mu$ by the Parseval identity of step 3.1, while the normalized torus measure and Lebesgue measure on $Q$ have the norm conversion ${\int }_Q\mathrm{\mid }E{\mathrm{\mid }}^2=L_1{L}_2{L}_3\mathrm{\parallel }E{\mathrm{\parallel }}_{L^2(\nu )}^2$ and likewise for $B$. Hence the energy $\frac{1}{2}{\int }_Q(\epsilon \mathrm{\mid }E{\mathrm{\mid }}^2+\mathrm{\mid }B{\mathrm{\mid }}^2\mathrm{/}\mu )dx$ of the constructed solution equals its initial value at every time. By the uniqueness of step 4.2, every $C(\mathbb{R};L^2)$ weak solution in the class with the same initial datum is this solution, so energy conservation holds for every weak solution.

The limit $(E,B)$ of step 4.1 is a weak solution in the sense of step 1.1. Indeed, each truncation $(E_M,B_M)$ solves the equations pointwise, hence also in the pairing formulation, because for smooth periodic fields the distributional pairings coincide with the integral pairings and the cur sign convention is the same. For a smooth periodic test pair $(\mathrm{\Psi },\mathrm{\Theta })$ compactly supported in $t$, the differences satisfy $\mathrm{\mid }\iint (E_M-E)\cdot {\mathrm{\partial }}_t\mathrm{\Psi }d\nu dt\mathrm{\mid }\le {\sup }_t\mathrm{\parallel }(E_M-E)(t){\mathrm{\parallel }}_{L^2}\cdot \int \mathrm{\parallel }{\mathrm{\partial }}_t\mathrm{\Psi }(t){\mathrm{\parallel }}_{L^2}dt$, and $\int \mathrm{\parallel }{\mathrm{\partial }}_t\mathrm{\Psi }(t)\mathrm{\parallel }dt<\mathrm{\infty }$ because $\mathrm{\Psi }$ is compactly supported in time and continuous, so the bound tends to $0$ by the uniform convergence of step 4.1; the pairings with $\mathrm{curl}\mathrm{\Psi }$ and $\mathrm{curl}\mathrm{\Theta }$, and the corresponding pairings of $B_M$, are handled identically. Passing to the limit in the two displayed identities therefore shows that $(E,B)$ is a weak solution. The divergence constraints hold in the limit because the class $\mathcal{C}$ is closed, as recorded in step 1.1.

Collecting the results: step 3.1 proves claim (i), namely that the finite separated Fourier modes are dense in the class and that the class is complete for the $L^2$ pairing; steps 4.1 and 5.2 construct, for every initial datum in the class, a $C(\mathbb{R};L^2)$ weak solution in the class, and step 4.2 proves that it is the only one, which is claim (ii); step 5.1 proves the energy constancy of claim (iii) for every weak solution. Countable Choice is used exactly through the torus-character, Fourier-convergence, Hilbert-space and regular-distribution suppliers of [L1] and [L4]; the finite-dimensional modal systems and the energy identity use no further choice. This proves every assertion of the statement.