Em Boundary And Spectral Prerequisites — Examples

The sphere trace and its harmonic extension. On $S^2$ the outward unit normal is $\nu (n)=n$, so the normal derivative of $x_3$ is ${\mathrm{\partial }}_n{x}_3=\mathrm{\nabla }{x}_3\cdot n=e_3\cdot n=n_3$; thus $x_3$ is harmonic on ${\mathbb{R}}^3$ by [F1] and has boundary trace $f(n)=n_3$ on $S^2$. The function $f$ is continuous, hence in $L^2(S^2)$ with coefficients $f_{lm}$, so the ball boundary branch [F7] applies: the series ${\sum }_{l,m}{r}^l{f}_{lm}{Y}_{lm}(n)$ is the unique harmonic function on $B_1$ with $L^2$ boundary limit $f$ and continuous extension to ${\bar{B}}_1$. Since $x_3$ is such a harmonic function, the branch identifies the extension as $u(x)=x_3$; in particular $f=n_3$ is an admissible ball-boundary datum, not an excluded class.

The total mass of the sphere measure. Apply the polar-coordinates identity [F3] with $n=3$ to the indicator of the closed unit ball, a Borel measurable function: ${\int }_{{\mathbb{R}}^3}{\mathbf{1}}_{{\bar{B}}_1}d{\lambda }_3={\int }_0^{\mathrm{\infty }}{\int }_{S^2}{\mathbf{1}}_{{\bar{B}}_1}(r\omega )r^{2}d\sigma (\omega )dr$; the left side is ${\lambda }_3({\bar{B}}_1)=4\pi \mathrm{/}3$ by [F4], and on the right side ${\mathbf{1}}_{{\bar{B}}_1}(r\omega )=1$ for $0\le r\le 1$ and every $\omega \in S^2$, so the iterated integral is $\left({\int }_0^1{r}^{2}dr\right)\sigma (S^2)=\sigma (S^2)\mathrm{/}3$; hence $\sigma (S^2)=3\cdot 4\pi \mathrm{/}3=4\pi$. The argument uses the Countable Choice assumption of the polar identity [F3].

The piecewise-linear function and its weak derivatives. Define $u:{\mathbb{R}}^3\to \mathbb{R}$ by $u(x)=x_1$ for $x_1<0$ and $u(x)=x_1\mathrm{/}2$ for $x_1>0$. Assign $u=0$ on the interface $\{x_1=0\}$; both branches tend to $0$ as $x_1\to 0$ through their sides, so $u$ is continuous on ${\mathbb{R}}^3$, and it is $C^{\mathrm{\infty }}$ on each of the two open half-spaces. Its distributional first derivatives are the locally integrable — indeed bounded — functions $g_1:={\mathbf{1}}_{\{x_1<0\}}+\frac{1}{2}{\mathbf{1}}_{\{x_1>0\}}$, $g_2=0$, $g_3=0$: for every $\phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$, integrating by parts on each half-space with the vanishing of the boundary terms (the traces $x_1\phi$ and $\frac{1}{2}{x}_1\phi$ vanish on $\{x_1=0\}$ and at infinity) gives $\int u{\mathrm{\partial }}_1\phi =-\int g_1\phi$, and the partial derivatives in the other two coordinates vanish identically on each side. Hence the weak first derivatives of $u$ are represented by $g_1,g_2,g_3$, so $u$ satisfies the weak-differentiability part of the definitions in [F6]; and $u$ is not $C^1$ on any neighbourhood of the interface, since its classical one-sided derivatives $1$ and $1\mathrm{/}2$ in the $x_1$ direction differ at every point of $\{x_1=0\}$.

The weak transmission equation. With $\epsilon =1$ on $\{x_1<0\}$ and $\epsilon =2$ on $\{x_1>0\}$, the field $\epsilon \mathrm{\nabla }u$ takes the value $(1,0,0)$ on both sides: on the first side $\epsilon {\mathrm{\partial }}_{1}u=1\cdot 1=1$ and on the second $\epsilon {\mathrm{\partial }}_{1}u=2\cdot \frac{1}{2}=1$, with the other components zero. Hence $\epsilon \mathrm{\nabla }u$ is the constant field $\mathbf{c}=(1,0,0)$, whose divergence is $0$ by [F2]; equivalently, for every $\phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$ one has $\int (\epsilon \mathrm{\nabla }u)\cdot \mathrm{\nabla }\phi dx=\int \mathbf{c}\cdot \mathrm{\nabla }\phi dx=0$, so the distributional equation $-\mathrm{div}(\epsilon \mathrm{\nabla }u)=0$ holds weakly on ${\mathbb{R}}^3$. Thus the weak formulation of the transmission problem is satisfied by a function that is not $C^1$, and its one-sided first derivatives jump from $1$ to $1\mathrm{/}2$ across the interface.

Incompatibility of the two sphere conditions. Suppose $w\in C^2({\bar{B}}_1)$ satisfied $\mathrm{\Delta }w=0$ in $B_1$ and ${\mathrm{\partial }}_{n}w=1$ on $S^2$, and apply the divergence theorem [F5] to $F=\mathrm{\nabla }w\in C^1({\bar{B}}_1;{\mathbb{R}}^3)$ on the bounded $C^1$ domain $B_1$: $0={\int }_{B_1}\mathrm{\Delta }wdx={\int }_{S^2}{\mathrm{\partial }}_{n}wd\sigma =\sigma (S^2)=4\pi$ by [F1] and step 1.2, a contradiction. Hence no such $w$ exists, so the conditions are incompatible; by step 1.1 the same trace $f=n_3$ is nevertheless an admissible ball-boundary datum whose harmonic extension has normal derivative $n_3$, not the constant $1$.

Scope of the elliptic conclusions. Step 2.1 produces a weak solution of $-\mathrm{div}(\epsilon \mathrm{\nabla }u)=0$ for a coefficient that is positive and smooth on each open side but discontinuous across the interface $\{x_1=0\}$, and step 1.3 shows the solution fails to be $C^1$ near the interface; this is a local smooth-interface witness with smooth coefficients on each closed side, and its one-sided regularity is explicit from the affine formulas. It illustrates the distinction made in [F8], but the global transmission theorem is not directly invoked: the plane is unbounded and is not a closed interface strictly inside a bounded domain as that theorem requires. Therefore a global $C^1$ or $C^{\mathrm{\infty }}$-on-the-closure conclusion for the piecewise-constant coefficient case is false, while the recorded one-sided piecewise regularity is the sharp conclusion; the smooth-coefficient statements of that theorem cannot be extended to this case.

Scope of the spectral completeness conclusions. The coefficient field of steps 1.3 and 2.1 is discontinuous across $\{x_1=0\}$ and its weak solution is not classical $C^1$ there, so the configuration is not one of the smooth classical configurations to which the smooth PEC cavity or rectangular classes of [F9] refer; that theorem declares its smooth-domain class and asserts in its statement that no completeness is asserted in non-smooth classes. The explicit weak solution of step 2.1 shows that the weak formulation of the equation remains valid in this excluded configuration and does not acquire the missing classical regularity, so it cannot be used to extend the spectral conclusions. This scalar transmission witness is not itself a vector PEC eigenfield and makes no claim that spectral completeness fails in any other correctly specified weak class; [F9] is used to inspect the stated scope, not to infer such a failure.

Audit. Step 1.1 identifies the admissible ball-boundary datum $f=n_3$ and its harmonic extension; step 1.2 computes the sphere mass $\sigma (S^2)=4\pi$ from the polar formula and the closed-ball volume; step 2.2 uses the divergence theorem to show that $\mathrm{\Delta }u=0$ in $B_1$ and ${\mathrm{\partial }}_{n}u=1$ on $S^2$ cannot hold simultaneously; steps 1.3 and 2.1 exhibit the continuous, weakly differentiable piecewise-linear solution with $\epsilon {\mathrm{\partial }}_{1}u\equiv 1$ and show that it is not $C^1$; and steps 3.1 and 3.2 place this witness against the exact hypotheses of the elliptic and spectral results. The elliptic and spectral citations [F8]-[F9] are used solely to compare their stated hypotheses and exclusions; their Axiom-of-Choice existence conclusions are not invoked here. Countable Choice is used through the ball-boundary supplier [F7], the polar-coordinate identity [F3] of step 1.2 and the divergence theorem [F5] of step 2.2, whose own assumptions are those of the Statement; no other choice principle is used. No measurement, physical field interpretation or empirical claim is made.