Weak fields, boundary power and dissipative uniqueness

Let $I\subseteq \mathbb{R}$ be a nonempty open interval and let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be a nonempty bounded Lipschitz polyhedron in the laboratory frame. Its boundary $\mathrm{\Gamma }=\mathrm{\partial }\mathrm{\Omega }$ has the outward unit normal $n$ on the relative interior of each planar face; edge and vertex values will be irrelevant under the face-area convention below. Use the SI conventions of this framework. In particular $E$ has units $\mathrm{V}{\mathrm{m}}^{-1}$, $H$ has units $\mathrm{A}{\mathrm{m}}^{-1}$, $D$ has units $\mathrm{C}{\mathrm{m}}^{-2}$, $B$ has units $\mathrm{T}$, and a volume current density $J$ has units $\mathrm{A}{\mathrm{m}}^{-2}$.

For the boundary integrals, let $\mathcal{F}$ be the finite family of relative interiors of the planar two-dimensional faces of $\mathrm{\Gamma }$. Write ${\lambda }_2$ for Lebesgue measure on ${\mathbb{R}}^2$, defined as the restriction of Lebesgue outer measure to its Carath'eodory-measurable sets; under the assumed ${\mathrm{A}\mathrm{C}}_{\omega }$, this is a complete measure. For each face $F\in \mathcal{F}$, let $P_F$ be its supporting plane and fix a Euclidean isometry ${\iota }_F:{\mathbb{R}}^2\to P_F$. For every Borel set $A\subseteq \mathrm{\Gamma }$ (a set in the sigma-algebra generated by the open subsets of $\mathrm{\Gamma }$), define $d{S}_F(A):={\lambda }_2({\iota }_F^{-1}(A\cap F))$. Any two choices of ${\iota }_F$ differ by an affine Euclidean isometry of ${\mathbb{R}}^2$ with absolute Jacobian determinant $1$, so the Borel change-of-variables formula makes $d{S}_F$ independent of that choice. Each face is bounded and lies in a finite rectangle in its plane, so $d{S}_F(F)<\mathrm{\infty }$. Set $dS:={\sum }_{F\in \mathcal{F}}d{S}_F$ on $\mathrm{\Gamma }$. The edges and vertices have $dS$-measure zero and are omitted, so $n$ is the outward face normal $dS$-almost everywhere; its values on edges and vertices do not affect any boundary expression. Since $dS$ is finite and Lebesgue measure restricted to $I$ is sigma-finite, $dS\otimes d{\lambda }_1{\mathrm{\mid }}_I$ is the unique sigma-finite product measure on $\mathrm{\Gamma }\times I$, with rectangle values $dS(A){\lambda }_1(J)$ for Borel $A\subseteq \mathrm{\Gamma }$ and Lebesgue-measurable $J\subseteq I$. All boundary $L^p(\mathrm{\Gamma })$ and $L^p(\mathrm{\Gamma }\times I)$ spaces and almost-everywhere statements use these measures.

Every cross product in the boundary formulas is the right-handed Euclidean cross product in the positively oriented orthonormal frame basis: for $u=(u_1,u_2,u_3)$ and $v=(v_1,v_2,v_3)$, $u\times v=(u_2{v}_3-u_3{v}_2,u_3{v}_1-u_1{v}_3,u_1{v}_2-u_2{v}_1)$, as in the cross-product definition of this library.

For a real vector field $u\in L^2(\mathrm{\Omega };{\mathbb{R}}^3)$, its distributional curl is the distribution defined by

$⟨\mathrm{curl}u,\phi ⟩={\int }_{\mathrm{\Omega }}u\cdot \mathrm{curl}\phi d{\lambda }_3,\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^3)\mathrm{.}$

As in the real graph space definition of the imported mathematical library, $H(\mathrm{curl};\mathrm{\Omega })$ consists of the $L^2$ vector fields whose distributional curl is represented by an $L^2$ vector field, with norm

$\mathrm{\parallel }u{\mathrm{\parallel }}_{H(\mathrm{curl};\mathrm{\Omega })}^2=\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2+\mathrm{\parallel }\mathrm{curl}u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2\mathrm{.}$

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ as carried by that imported weak-derivative convention. Let $\epsilon ,\mu :\mathrm{\Omega }\to {\mathbb{R}}^{3\times 3}$ be measurable, essentially bounded, real symmetric coefficient fields, independent of time. Require constants $0<a_{\epsilon }\le b_{\epsilon }<\mathrm{\infty }$ and $0<a_{\mu }\le b_{\mu }<\mathrm{\infty }$ such that, for almost every $x\in \mathrm{\Omega }$ and every $v\in {\mathbb{R}}^3$,

$a_{\epsilon }\mathrm{\mid }v{\mathrm{\mid }}^2\le v\cdot \epsilon (x)v\le b_{\epsilon }\mathrm{\mid }v{\mathrm{\mid }}^2,a_{\mu }\mathrm{\mid }v{\mathrm{\mid }}^2\le v\cdot \mu (x)v\le b_{\mu }\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{.}$

The coefficient units are $\mathrm{F}{\mathrm{m}}^{-1}$ and $\mathrm{H}{\mathrm{m}}^{-1}$, respectively. They define bounded symmetric multiplication operators on $L^2$. Define $D(t)=\epsilon E(t)$ and $B(t)=\mu H(t)$. The spatially varying coefficient fields are explicit local model data; they are not inferred from the homogeneous material-closure postulate. Its constant symmetric branch is a special case.

A field $E:I\to L^2(\mathrm{\Omega };{\mathbb{R}}^3)$ is in $C^1(I;L^2)$ when it has a derivative in the $L^2$ norm at every $t\in I$ and that derivative is norm-continuous. $C(I;H(\mathrm{curl};\mathrm{\Omega }))$ means norm-continuity into the graph space. A finite-energy model solution is a pair

$E,H\in C^1(I;L^2(\mathrm{\Omega };{\mathbb{R}}^3))\cap C(I;H(\mathrm{curl};\mathrm{\Omega })),$

with $J\in C(I;L^2(\mathrm{\Omega };{\mathbb{R}}^3))$, satisfying the adopted evolution equations as $L^2$ identities for each $t\in I$,

$\epsilon {\mathrm{\partial }}_{t}E=\mathrm{curl}H-J,\mu {\mathrm{\partial }}_{t}H=-\mathrm{curl}E\mathrm{.}$

This class specifies the Maxwell curl-evolution subsystem independently of charge data. In the volume-charge case, a full Maxwell application additionally supplies a charge density $\rho (\cdot ,t)\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$ for every $t\in I$, in $\mathrm{C}{\mathrm{m}}^{-3}$, and imposes the Gauss constraints: for every $t\in I$ and every scalar test $\psi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$,

${\int }_{\mathrm{\Omega }}B(t)\cdot \mathrm{\nabla }\psi d{\lambda }_3=0,-{\int }_{\mathrm{\Omega }}D(t)\cdot \mathrm{\nabla }\psi d{\lambda }_3={\int }_{\mathrm{\Omega }}\rho (\cdot ,t)\psi d{\lambda }_3\mathrm{.}$

These additional constraints are not part of the evolution class defined here and are not inferred from its two curl equations. Energy identities for this class also apply to its Gauss-constrained subclass; they do not establish the constraints or the physical applicability of either class.

Its stored field energy is the finite real number

$W(t):=\frac{1}{2}{\int }_{\mathrm{\Omega }}(E(x,t)\cdot \epsilon (x)E(x,t)+H(x,t)\cdot \mu (x)H(x,t))d{\lambda }_3(x)\mathrm{.}$

There are two alternative boundary models.

PEC. Perfect electric conductor means $E(t)\in H_0(\mathrm{curl};\mathrm{\Omega })$ for every $t$, where $H_0(\mathrm{curl};\mathrm{\Omega })$ is exactly the closure of $C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^3)$ in the displayed graph norm. This closure definition is used; it is not replaced by an unproved identification with every possible trace kernel.

Impedance. For $u\in H(\mathrm{curl};\mathrm{\Omega })$, an $L^2$ weak tangential trace is a field $u_{\tau }\in L^2(\mathrm{\Gamma };{\mathbb{R}}^3)$ with $n\cdot u_{\tau }=0$ almost everywhere such that

${\int }_{\mathrm{\Omega }}(\mathrm{curl}u\cdot \phi -u\cdot \mathrm{curl}\phi )d{\lambda }_3={\int }_{\mathrm{\Gamma }}(n\times u_{\tau })\cdot \phi {\mathrm{\mid }}_{\mathrm{\Gamma }}dS\text{for every }\phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3;{\mathbb{R}}^3)\mathrm{.}$

Here a test field $\phi$ is a smooth map on all of ${\mathbb{R}}^3$ with compact support, and $\phi {\mathrm{\mid }}_{\mathrm{\Gamma }}$ is its ordinary restriction. In particular, tests may be nonzero on the boundary. This identity identifies the boundary field through independently chosen tests; the single energy-pair identity below does not replace it.

A Green-compatible tangential boundary pair for $E,H$ consists of measurable fields $E_{\tau },H_{\tau }\in L^2(\mathrm{\Gamma }\times I;{\mathbb{R}}^3)$ such that, outside a single set of times of Lebesgue measure zero, $E_{\tau }(\cdot ,t)$ and $H_{\tau }(\cdot ,t)$ are weak tangential traces of $E(t)$ and $H(t)$, respectively, in the displayed sense, and additionally

${\int }_{\mathrm{\Omega }}(\mathrm{curl}H\cdot E-H\cdot \mathrm{curl}E)d{\lambda }_3=-{\int }_{\mathrm{\Gamma }}n\cdot (E_{\tau }\times H_{\tau })dS\mathrm{.}$

The energy-pair Green identity is retained as an explicit admissibility requirement; no extension from smooth tests to the rough fields themselves is assumed without it. Membership in $H(\mathrm{curl})$ alone does not assert that such an $L^2$ trace pair exists. Let $Y\in L^{\mathrm{\infty }}(\mathrm{\Gamma })$ be a time-independent scalar surface admittance, in siemens, with $Y\ge 0$ almost everywhere. The impedance law is

$H_{\tau }=Y(n\times E_{\tau })\text{almost everywhere on }\mathrm{\Gamma }\times I\mathrm{.}$

Then $Y\mathrm{\mid }{E}_{\tau }{\mathrm{\mid }}^2\in L^1(\mathrm{\Gamma }\times I)$ and is the nonnegative outward boundary-power density in $\mathrm{W}{\mathrm{m}}^{-2}$. No pointwise boundary value or existence of an $L^2$ weak tangential trace for an arbitrary $H(\mathrm{curl})$ field is asserted.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The cross product in ${\mathbb{R}}^3$

Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$

Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields

The product measure of two sigma-finite measure spaces

Framework-local SI units, the vacuum constants and the exact/measured status of their values

The real H(curl) graph space on a polyhedron

Zero tangential trace H(curl) space

Borel change of variables from the compact-support formula and Radon uniqueness

Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch

Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume

For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique

Assume the surface-measure convention with ${\mathrm{A}\mathrm{C}}_{\omega }$ of the mathematical definition of the stationary surface distribution. Let $S\subset {\mathbb{R}}^3$ be a compact embedded $C^2$ hypersurface without manifold boundary, with a chosen continuous unit normal $n$ and its finite surface measure $dS$. At $x\in S$, define the tangent plane

$T_{x}S:=\{v\in {\mathbb{R}}^3:v\cdot n(x)=0\}\mathrm{.}$

For $\varphi \in C^1(S)$, meaning that its composition with each local $C^2$ surface chart is $C^1$, its surface gradient ${\mathrm{\nabla }}_S\varphi (x)\in T_{x}S$ is the unique vector satisfying

$v\cdot {\mathrm{\nabla }}_S\varphi (x)=d{\varphi }_x(v)\text{for every }v\in T_{x}S,$

where $d{\varphi }_x$ is the derivative of $\varphi$ in tangent directions. For a measurable tangential field $K:S\to {\mathbb{R}}^3$ with ${\int }_S\mathrm{\mid }K\mathrm{\mid }dS<\mathrm{\infty }$, define its weak surface divergence ${\mathrm{Div}}_{S}K$ as the linear functional on $C^1(S)$

$⟨{\mathrm{Div}}_{S}K,\varphi ⟩:=-{\int }_{S}K\cdot {\mathrm{\nabla }}_S\varphi dS\mathrm{.}$

This functional is well-defined because $S$ is compact, ${\mathrm{\nabla }}_S\varphi$ is continuous and bounded, and $K$ is integrable. If it has a continuous density $g:S\to \mathbb{R}$, meaning

$⟨{\mathrm{Div}}_{S}K,\varphi ⟩={\int }_{S}g\varphi dS\text{for every }\varphi \in C^1(S),$

then $g$ is called a continuous representative of the weak surface divergence. The definition itself does not assert that such a representative exists or is unique; a result using pointwise ${\mathrm{div}}_{S}K$ must establish both.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Surface distributions and interfacial jumps

The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$

Use the finite-energy boundary model defined earlier on the nonempty open time interval $I$ and bounded Lipschitz polyhedron $\mathrm{\Omega }$. Thus $\epsilon ,\mu$ are time-independent real symmetric essentially bounded coefficient fields with uniform positive lower bounds, $E,H\in C^1(I;L^2(\mathrm{\Omega };{\mathbb{R}}^3))\cap C(I;H(\mathrm{curl};\mathrm{\Omega }))$, $J\in C(I;L^2(\mathrm{\Omega };{\mathbb{R}}^3))$, and

$\epsilon {\mathrm{\partial }}_{t}E=\mathrm{curl}H-J,\mu {\mathrm{\partial }}_{t}H=-\mathrm{curl}E$

hold as $L^2$ identities for every $t\in I$. Define

$W(t):=\frac{1}{2}{\int }_{\mathrm{\Omega }}(E\cdot \epsilon E+H\cdot \mu H)d{\lambda }_3\mathrm{.}$

For every $s<t$ in $I$, the closure-defined PEC branch satisfies

$W(t)-W(s)=-{\int }_s^t{\int }_{\mathrm{\Omega }}E\cdot Jd{\lambda }_{3}dr\mathrm{.}$

For the Green-compatible impedance branch, where $H_{\tau }=Y(n\times E_{\tau })$, it satisfies

$W(t)-W(s)=-{\int }_s^t{\int }_{\mathrm{\Omega }}E\cdot Jd{\lambda }_{3}dr-{\int }_s^t{\int }_{\mathrm{\partial }\mathrm{\Omega }}Y\mathrm{\mid }{E}_{\tau }{\mathrm{\mid }}^{2}dSdr\mathrm{.}$

Here $E_{\tau },H_{\tau }$ and the Green identity are exactly the $L^2$ boundary data and admissibility condition in the model definition; no trace is inferred from H(curl) membership alone. The work density $E\cdot J$ has SI units $\mathrm{W}{\mathrm{m}}^{-3}$; the boundary term has units $\mathrm{W}{\mathrm{m}}^{-2}$ per unit area and is nonnegative in the impedance branch. If $J=0$ and $E(t_0)=H(t_0)=0$ in $L^2$, the only admissible solution is zero for $t\ge t_0$. For two solutions with the same coefficients, current and initial data, forward equality follows in the PEC branch. In the impedance branch it follows if the difference fields, with the difference boundary pair and the same $Y$, also satisfy the model's Green compatibility identity. This difference-pair identity is an explicit hypothesis here; no inference from the two separate energy-pair identities alone is used. These conclusions concern the curl-evolution class and infer no Gauss constraint. Uniqueness is forward from the specified initial time; no backward uniqueness is asserted for a dissipative boundary.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $S\subset {\mathbb{R}}^3$ be a compact embedded $C^2$ hypersurface without manifold boundary, with surface measure $dS$ and a chosen unit normal. Let $K$ be a $C^1$ tangential vector field on $S$, meaning that its two coordinate components in every $C^2$ surface chart are $C^1$. Then the weak surface divergence ${\mathrm{Div}}_{S}K$ defined earlier has a unique continuous density $g:S\to \mathbb{R}$ such that

$⟨{\mathrm{Div}}_{S}K,\varphi ⟩={\int }_{S}g\varphi dS\text{for every }\varphi \in C^1(S)\mathrm{.}$

In a chart $X:V\subseteq {\mathbb{R}}^2\to S$, write $K=K^1{\mathrm{\partial }}_{1}X+K^2{\mathrm{\partial }}_{2}X$ and let $J_X=\sqrt{\det ({\mathrm{\partial }}_{a}X\cdot {\mathrm{\partial }}_{b}X{)}_{a,b=1}^2}$ be its surface-area density. The representative satisfies

$g(X(u))=J_X(u{)}^{-1}{\sum }_{a=1}^2{\mathrm{\partial }}_a(J_X(u)K^a(u))\mathrm{.}$

This unique continuous representative is denoted ${\mathrm{div}}_{S}K$.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $U=I\times \mathrm{\Omega }$ be a laboratory region, where $I\subseteq \mathbb{R}$ is a nonempty open interval and $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ is open. Use the field and SI conventions of The electric and magnetic fields as fixed-time spatial vector fields, Polarization, magnetization, bound and free sources, and the fields D and H, and Framework-local SI units, the vacuum constants and the exact/measured status of their values: $E$ is measured in $\mathrm{V}{\mathrm{m}}^{-1}$, $B$ in tesla, $D$ in $\mathrm{C}{\mathrm{m}}^{-2}$, and $H$ in $\mathrm{A}{\mathrm{m}}^{-1}$. Let $S\subset \mathrm{\Omega }$ be a fixed compact two-sided embedded $C^2$ hypersurface without manifold boundary, with unit normal $n$ pointing from the minus side ${\mathrm{\Omega }}_{-}$ to the plus side ${\mathrm{\Omega }}_{+}$; write $[F]=F_{+}-F_{-}$. Let $E,B,D,H$ be locally integrable fields with the joint one-sided spacetime $C^1$ extensions on the two sides used in the stationary-interface theorem. Let the volume charge ${\rho }_v$ and volume current $J_v$ be locally integrable and $C^1$ on each side with joint one-sided $C^1$ extensions to $S\times I$. Let $\sigma :S\times I\to \mathbb{R}$ and $K:S\times I\to {\mathbb{R}}^3$ be $C^1$, with $K\cdot n=0$. Their SI units are $\mathrm{C}{\mathrm{m}}^{-3}$ for ${\rho }_v$, $\mathrm{A}{\mathrm{m}}^{-2}$ for $J_v$, $\mathrm{C}{\mathrm{m}}^{-2}$ for $\sigma$, and $\mathrm{A}{\mathrm{m}}^{-1}$ for $K$.

Assume the distributional macroscopic equations with the stationary sheet source pair ${\rho }_f={\rho }_v+\sigma {\delta }_S$, $J_f=J_v+K{\delta }_S$:

$\mathrm{div}D={\rho }_f,\mathrm{div}B=0,\mathrm{curl}E=-{\mathrm{\partial }}_{t}B,\mathrm{curl}H=J_f+{\mathrm{\partial }}_{t}D$

on $U$. This is the thin-interface solution assumption of the adopted macroscopic model; it is not inferred from the smooth equations.

Then pointwise on $S\times I$,

$n\cdot [D]=\sigma ,n\cdot [B]=0,n\times [E]=0,n\times [H]=K,$

and

${\mathrm{\partial }}_t\sigma +{\mathrm{div}}_{S}K+n\cdot [J_v]=0.$

Here ${\mathrm{Div}}_{S}K$ is the weak surface divergence of Weak surface divergence of a tangential field. The proof uses Continuous density of the weak surface divergence for a C1 tangent field to identify its unique continuous density for the stated $C^1$ current; ${\mathrm{div}}_{S}K$ denotes that density. The sign of $n\cdot [J_v]$ uses $[J_v]=J_{v,+}-J_{v,-}$ for the chosen normal. In the stationary-sheet vacuum model, if there is no point-source component in the interface neighbourhood and $D={\epsilon }_{0}E,H=B\mathrm{/}{\mu }_0$, the adopted distributional vacuum Maxwell equations supply the displayed macroscopic equations, so this theorem applies there as well.