Stationary macroscopic interface jump conditions for D, B, E and H

The surface $S$ is Lebesgue-null in $\mathrm{\Omega }$: by [L3] the finitely many regular charts $X_j:V_j\to S$ cover $S$, each image $X_j(V_j)$ is null in ${\mathbb{R}}^3$ by [L5] because $X_j$ is a $C^1$ map from the lower-dimensional open set $V_j\subseteq {\mathbb{R}}^2$, and a finite union of null sets is null by [L5]. Consequently, for $F\in \{B,D\}$ the distributional time derivative of the regular distribution of $F$ is the regular distribution of the piecewise classical time derivative: fix a test $\psi$ with support in a compact cylinder $\bar{Q}\times J\subseteq U$, so that $⟨{\mathrm{\partial }}_{t}F,\psi ⟩=-{\int }_{U}F{\mathrm{\partial }}_t\psi$; by [L8] applied after Fubini, this iterated integral equals $-{\int }_Q{\int }_{J}F(x,t){\mathrm{\partial }}_t\psi (x,t)dtdx$, and for every $x\in \mathrm{\Omega }\setminus S$ the inner integrand is $C^1$ in $t$ with $\psi (x,\cdot )$ compactly supported in $J$, so [L8] gives ${\int }_{J}F(x,t){\mathrm{\partial }}_t\psi (x,t)dt=-{\int }_J{\mathrm{\partial }}_{t}F(x,t)\psi (x,t)dt$; since $S$ is null, replacing the outer integral over $\mathrm{\Omega }\setminus S$ by one over $\mathrm{\Omega }$ changes nothing, so $⟨{\mathrm{\partial }}_{t}F,\psi ⟩={\int }_U\{{\mathrm{\partial }}_{t}F\}\psi$ for every test $\psi$, where $\{{\mathrm{\partial }}_{t}F\}$ is locally bounded by the joint one-sided $C^1$ extension hypothesis. In the weak-trace branch, the assumed $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ temporal derivatives instead make the same compact-test products integrable, so the identical Fubini and line-integration argument applies.

By [L1], applied to $D$, $B$, $E$, $H$ (each is $C^1$ on ${\mathrm{\Omega }}_{-}\cup {\mathrm{\Omega }}_{+}$ with one-sided $C^1$ traces under the joint extension hypothesis), the four spatial identities $\mathrm{div}D=\{\mathrm{div}D\}+n\cdot [D]{\delta }_S$, $\mathrm{div}B=\{\mathrm{div}B\}+n\cdot [B]{\delta }_S$, $\mathrm{curl}E=\{\mathrm{curl}E\}+n\times [E]{\delta }_S$ and $\mathrm{curl}H=\{\mathrm{curl}H\}+n\times [H]{\delta }_S$ hold as distributions on $U$ under ${\mathrm{A}\mathrm{C}}_{\omega }$, with the piecewise fields locally bounded near $S$. More explicitly, apply [L1] at each fixed time to $\psi (\cdot ,t)$ and integrate its identity in $t$. Fubini is justified by the compact-test integrability of the fields, their piecewise spatial derivatives and their surface-time traces; in the weak-trace branch these are exactly the additional stated $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ hypotheses, so the same transfer to spacetime distributions applies without a uniform bound.

[Coefficient extraction] Let $g\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(U)$ and let $a$ be measurable and locally integrable on $S\times I$ for $dSdt$, with $u_g+a{\delta }_S=0$. Every smooth test supported away from $S\times I$ has zero sheet pairing; the fundamental lemma [L7] applied on that open complement therefore gives $g=0$ almost everywhere there. Since $S$ is null by step 1.1, Fubini [L8] makes $S\times I$ null locally in spacetime, so $g=0$ almost everywhere on $U$. Thus ${\int }_{S\times I}a\psi dSdt=0$ for every ambient smooth compact test $\psi$. In a sufficiently small rigid graph chart write $X(y)=(y,h(y))$, with a product box $V\times (z_{-},z_{+})\times J$ contained in the ambient chart/time domain. Shrink $V$ and the height interval so the graph over $V$ stays in a closed subinterval inside $(z_{-},z_{+})$. For any $\eta \in C_c^{\mathrm{\infty }}(V\times J)$ choose $\beta \in C_c^{\mathrm{\infty }}((z_{-},z_{+}))$ equal to $1$ on that graph-height interval by [L6], and take the ambient smooth test $\psi (y,z,t)=\eta (y,t)\beta (z)$, transported by the rigid motion and extended by zero. This test uses no composition with the merely $C^2$ graph function; its restriction to $S$ is $\eta$. The chart formula [L3] gives ${\int }_{V\times J}(a\circ X)(y,t)J_X(y)\eta (y,t)dydt=0$. By [L7], $(a\circ X)J_X=0$ almost everywhere; positivity and continuity of $J_X$ give $a=0$ surface-time almost everywhere. Such small boxes cover every chart, so the conclusion holds on $S\times I$. In the joint branch $a\circ X$ is continuous; if it were nonzero at one point it would stay bounded away from zero on a nonempty parameter-time box of positive Lebesgue measure, contradicting the almost-everywhere conclusion. Thus $a=0$ pointwise in that branch.

Substituting the spatial identities of step 1.2 and the temporal identity of step 1.1 into the assumed identities [A1] gives four distributional identities of the form $u_g+a{\delta }_S=0$ with locally integrable volume parts: $u_{\{\mathrm{div}D\}-{\rho }_v}+(n\cdot [D]-{\sigma }_f){\delta }_S=0$, $u_{\{\mathrm{div}B\}}+n\cdot [B]{\delta }_S=0$, $u_{\{\mathrm{curl}E\}+\{{\mathrm{\partial }}_{t}B\}}+n\times [E]{\delta }_S=0$ and $u_{\{\mathrm{curl}H\}-J_v-\{{\mathrm{\partial }}_{t}D\}}+(n\times [H]-K_f){\delta }_S=0$; the sums are locally integrable because the piecewise fields and $J_v,{\rho }_v$ are locally integrable and the piecewise derivatives are locally bounded near $S$ (joint branch) or assumed locally integrable (weak-trace branch), and the sheet coefficients are the stated one-sided jump densities, continuous in the joint branch.

Apply step 2.1 to each of the four identities of step 2.2: the four coefficients $n\cdot [D]-{\sigma }_f$, $n\cdot [B]$, $n\times [E]$ and $n\times [H]-K_f$ vanish $dSdt$-almost everywhere, and in the joint branch they are continuous on $S\times I$ (one-sided traces of locally uniformly continuous extensions), hence vanish pointwise on $S\times I$; this proves $n\cdot [D]={\sigma }_f$, $n\cdot [B]=0$, $n\times [E]=0$ and $n\times [H]=K_f$ pointwise, equivalently ${\sigma }_f=n\cdot [D]$ and $K_f=n\times [H]$, and the inner and cross products are those of the one-sided traces written in the frame basis of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields. In the weak-trace branch the four coefficients vanish $dSdt$-almost everywhere; since the traces are spatially continuous at each fixed time, vanishing on a set of full $dS$-measure for almost every $t$ upgrades to every $x\in S$ for almost every $t$, and if the jump densities are additionally jointly continuous, to every $(x,t)\in S\times I$; the four identities hold as identities of sheet distributions in both branches, and all statements are conditional on the assumed distributional equations [A1] with ${\mathrm{A}\mathrm{C}}_{\omega }$ carried from step 1.2.