Weak fields, boundary power and dissipative uniqueness — Examples

Every distributional vacuum Maxwell solution with a point-charge source has a locally integrable ordinary electric-energy density $u_E=\frac{{\epsilon }_0}{2}\mathrm{\mid }E{\mathrm{\mid }}^2$ near the charge.

Use the boundary geometry of the preceding finite-energy model, with $\mathrm{\Gamma }=\mathrm{\partial }\mathrm{\Omega }$. On a flat boundary face with outward normal $n=e_z$, take a right-handed tangent basis $e_x,e_y$, with $e_z\times e_x=e_y$, and prescribe tangential boundary data $E_{\tau }=E_0{e}_x$ for $E_0\in \mathbb{R}$. For surface admittance $Y\ge 0$, the boundary condition $H_{\tau }=Y(n\times E_{\tau })$ gives $H_{\tau }=Y{E}_0{e}_y$ and outward power

$n\cdot (E_{\tau }\times H_{\tau })=Y{E}_0^2\ge 0.$

If the sign in the boundary law is reversed, $H_{\tau }=-Y(n\times E_{\tau })$, the outward power is $-Y{E}_0^2$; it is strictly inward when $Y{E}_0^2>0$.

For the conditional energy consequence, suppose an actual finite-energy model solution exists on a nonempty open interval $I$ and bounded Lipschitz polyhedron $\mathrm{\Omega }$: $E,H\in C^1(I;L^2(\mathrm{\Omega };{\mathbb{R}}^3))\cap C(I;H(\mathrm{curl};\mathrm{\Omega }))$, $J\in C(I;L^2(\mathrm{\Omega };{\mathbb{R}}^3))$, the two evolution equations of the preceding boundary model hold as $L^2$ identities, and tangential boundary fields $E_{\tau },H_{\tau }\in L^2(\mathrm{\Gamma }\times I;{\mathbb{R}}^3)$ satisfy its Green compatibility identity. Take $Y\in L^{\mathrm{\infty }}(\mathrm{\Gamma })$, $Y\ge 0$, as in that model. If the reversed law $H_{\tau }=-Y(n\times E_{\tau })$ holds almost everywhere on all of $\mathrm{\Gamma }\times I$, then the model energy $W(t)=\frac{1}{2}{\int }_{\mathrm{\Omega }}(E\cdot \epsilon E+H\cdot \mu H)d{\lambda }_3$ obeys

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

In particular, if $J=0$ and the last integral is positive, the stored energy increases. This conclusion is conditional on existence of a solution satisfying the reversed law.

Finite-energy PEC and impedance boundary formulation

Energy balance and zero-data forward uniqueness for admissible weak Maxwell fields

Chen's boundary relation is $n\times H-\lambda E_t=g$. Here $E_t$ denotes the tangential electric component, identified with $E_{\tau }$, and $g$ is prescribed tangential boundary data. For comparison with the homogeneous law of this example, set $g=0$ and use $H_{\tau }=Y(n\times E_{\tau })$. Since $\mathrm{\mid }n\mathrm{\mid }=1$ and $n\cdot E_{\tau }=0$, the right-handed cross product gives $n\times H_{\tau }=Yn\times (n\times E_{\tau })=-Y{E}_{\tau }$. Thus our law corresponds to Chen's homogeneous relation with $\lambda =-Y$. This is a conversion of conventions; the cited passage does not stipulate a sign for $\lambda$ or supply a passivity claim.

The drawing is a projection of the tangent plane: $e_z$ points out of the page, and the right-handed convention is $e_x\times e_y=e_z$. The arrows depict the representative case $E_0>0$ and $Y>0$. If $E_0<0$ while $Y>0$, both tangential arrows reverse; if $E_0=0$, both vectors vanish; and if $Y=0$, $H_{\tau }$ vanishes. The verification below supplies the sign calculation.

\begin{tikzpicture}[>=stealth,scale=1.1]
  \draw[->] (-0.2,0) -- (2.8,0) node[below] {$e_x$};
  \draw[->] (0,-0.2) -- (0,2.5) node[left] {$e_y$};
  \draw[->,thick,blue] (0,0) -- (1.7,0) node[below=4pt] {$E_\tau=E_0e_x$};
  \draw[->,thick,red] (0,0) -- (0,1.7) node[left=4pt] {$H_\tau=YE_0e_y$};
  \node at (2.2,1.4) {$n=e_z\ \odot$};
\end{tikzpicture}