Smooth boundary regularity of PEC cavity eigenfields in the declared graph completion

Polarizing the Gaffney identity [F1] gives, for smooth PEC $u,v$, ${\int }_{\mathrm{\Omega }}(\mathrm{curl}u\cdot \mathrm{curl}v+\mathrm{div}u\mathrm{div}v)={\int }_{\mathrm{\Omega }}\mathrm{\nabla }u:\mathrm{\nabla }v+{\int }_{\mathrm{\partial }\mathrm{\Omega }}\kappa u_n{v}_{n}dS$: the integration-by-parts boundary term before imposing the traces is ${\sum }_{i,j}{\int }_{\mathrm{\partial }\mathrm{\Omega }}({\nu }_j{u}_j{\mathrm{\partial }}_i{v}_i-{\nu }_j{u}_i{\mathrm{\partial }}_i{v}_j)$, and at the boundary $u=u_n\nu$, $v=v_n\nu$ with $\nu \cdot {\mathrm{\partial }}_i\nu =0$, so it reduces to $\kappa u_n{v}_n$; the sign is positive for an outward sphere with $\kappa =2\mathrm{/}R$, and no convexity is assumed.

By the $H^1$ equivalence and bounded trace of [F1] the identity of step 1.1 extends to all $u,v\in X$; combining it with $\mathrm{div}u=0$ and the eigenidentity of the Statement gives the weak mixed boundary identity ${\int }_{\mathrm{\Omega }}\mathrm{\nabla }u:\mathrm{\nabla }v+{\int }_{\mathrm{\partial }\mathrm{\Omega }}\kappa u_n{v}_{n}dS=\lambda {\int }_{\mathrm{\Omega }}u\cdot v$ for every $v\in X$; no classical normal derivative has been assumed.

Flattening: shrink smooth boundary charts and take a finite cover of the compact boundary with nested half-boxes $(y_1,y_2,s)$, $s>0$ inside, and cutoffs away from artificial sides; choose a smooth orthonormal frame $Q=({\tau }_1,{\tau }_2,\nu )$ with ${\tau }_2=\nu \times {\tau }_1$ and write $u=QU$. The $H^1$ trace has $T{U}_1=T{U}_2=0$ while $T{U}_3$ is free, and by [F2] every localized $H^1$ triple with these two vanishing traces is an admissible test for the identity of step 2.1 after odd/even extension, mollification and cut off; For completeness, for a localized scalar $z$ the odd extension is $z(y,s)$ for $s>0$ and $-z(y,-s)$ for $s<0$, while the even extension uses $z(y,-s)$ on $s<0$. The even extension has matching traces for every $z\in H^1$; the odd one has matching traces when $Tz=0$. On almost every normal line the ACL representative has a finite endpoint limit, because its derivative is $L^2$ on the bounded interval and hence $L^1$; this limit is the flat chart trace. Integration along those lines and then Fubini [F2] show that the weak normal derivative is the reflected derivative with the corresponding sign: the only possible boundary jump term is the difference of the two endpoint traces, which vanishes. Tangential differentiation acts on each half independently, so all reflected weak derivatives lie in $L^2$. Take a nonnegative smooth compact mollifier even in $s$, obtained by symmetrizing a cutoff from [F2] and normalizing its positive integral. Convolution preserves odd/even parity and converges together with all first weak derivatives in $L^2$ by [F2]; multiplication by an even cutoff away from the artificial sides preserves the parity. The resulting smooth first two components vanish at $s=0$, and the third has unrestricted value there. Pulling back through the smooth frame and chart and extending by zero outside the compact chart support therefore gives actual smooth PEC fields whose $H^1$ limits are the localized test triples, proving that those tests belong to $X$.

Local coupled weak system: expanding $\mathrm{\nabla }(QU)$ in step 2.1 and transporting the volume and surface measures produces a real symmetric smooth principal matrix $a^{ij}$, uniformly positive on the smaller half-box and scalar in the component indices, such that for all admissible triples $V$ ${\sum }_m\int a^{ij}{\mathrm{\partial }}_j{U}_m{\mathrm{\partial }}_i{V}_m+{\int }_B\beta U_3{V}_3=\int f\cdot V+{\sum }_i\int G_i\cdot {\mathrm{\partial }}_{i}V+{\int }_{B}h{V}_3$, where $B=\{s=0\}$, $\beta =\kappa J_B$ is smooth and possibly signed, $f=\lambda JU+{\sum }_i{C}_i{\mathrm{\partial }}_{i}U+C_{0}U\in L^2$, $G_i=B_{i}U\in H^1$ for smooth matrix coefficients, and $h=0$ in the original eigenidentity; the estimate below also allows an additional datum $h\in H^1(B)$. This follows by expanding the finite products, the two differentiated factors producing the scalar $a^{ij}$ because $Q^{\mathsf{T}}Q=I$, while terms with derivatives of $Q$ and $U$ enter $f$ or $G$, with no extra boundary datum.

Tangential mixed estimate: for $k=1,2$ put $W={\delta }_h^{k}U$ and test with $V=-{\delta }_{-h}^k({\eta }^{2}W)$ on a nested patch; the test is admissible by [F2], discrete integration by parts and the shifted product rule give the principal term ${\sum }_m\int {\eta }^2{a}^{ij}(x+h{e}_k){\mathrm{\partial }}_j{W}_m{\mathrm{\partial }}_i{W}_m\ge \alpha \mathrm{\parallel }\eta \mathrm{\nabla }W{\mathrm{\parallel }}_2^2$, and all remaining terms involve ${\delta }_{h}a$, $\mathrm{\nabla }\eta$, $\mathrm{\nabla }U$, $W$ and the Robin product; [F3] bounds them by $ϵ\mathrm{\parallel }\eta \mathrm{\nabla }W{\mathrm{\parallel }}_2^2+C_{ϵ}\mathrm{\parallel }U{\mathrm{\parallel }}_{H^1}^2$ on the larger patch using $\mathrm{\parallel }W{\mathrm{\parallel }}_2\le C\mathrm{\parallel }{\mathrm{\partial }}_{k}U{\mathrm{\parallel }}_2$ and $\mathrm{\parallel }{\delta }_{-h}({\eta }^{2}W){\mathrm{\parallel }}_2\le C(\mathrm{\parallel }\eta \mathrm{\nabla }W{\mathrm{\parallel }}_2+\mathrm{\parallel }W{\mathrm{\parallel }}_2)$ from the elementary line formula and Cauchy-Schwarz, the $f$ pairing is bounded without differentiating $f$ using the displayed bound for the test in $L^2$; transfer the quotient in the $G$ pairing to $G$ and use $\mathrm{\parallel }{\delta }_{h}G{\mathrm{\parallel }}_2\le C\mathrm{\parallel }G{\mathrm{\parallel }}_{H^1}$, and transfer the boundary quotient to $h$ using $\mathrm{\parallel }{\delta }_{h}h{\mathrm{\parallel }}_{L^2(B)}\le C\mathrm{\parallel }h{\mathrm{\parallel }}_{H^1(B)}$. The exact Robin quotient is ${\int }_B\beta (y+h){\eta }^2\mathrm{\mid }{W}_3{\mathrm{\mid }}^2+{\int }_B({\delta }_h\beta )U_3{\eta }^2{W}_3$. The trace estimate of [F1] $\mathrm{\parallel }Tz{\mathrm{\parallel }}_2^2\le ϵ\mathrm{\parallel }\mathrm{\nabla }z{\mathrm{\parallel }}_2^2+C_{ϵ}\mathrm{\parallel }z{\mathrm{\parallel }}_2^2$ absorbing the Robin and boundary-datum terms even when $\beta$ has either sign; choosing the finitely many epsilons small gives $\mathrm{\parallel }\eta \mathrm{\nabla }{\delta }_h^{k}U{\mathrm{\parallel }}_2\le C(\mathrm{\parallel }U{\mathrm{\parallel }}_{H^1}+\mathrm{\parallel }f{\mathrm{\parallel }}_2+\mathrm{\parallel }G{\mathrm{\parallel }}_{H^1}+\mathrm{\parallel }h{\mathrm{\parallel }}_{H^1(B)})$ uniformly for small $h\mathrm{\ne }0$.

Normal recovery: by step 5.1 each distributional tangential derivative of every ${\mathrm{\partial }}_i{U}_m$ is an $L^2$ class, since its pairing with a smooth test is the limit of $\int {\delta }_h^k({\mathrm{\partial }}_i{U}_m)\psi$ and is a bounded $L^2$ functional, represented by [F3]'s Riesz theorem; interior quotient tests give the same conclusion away from the boundary; hence the interior equation $-{\mathrm{\partial }}_i(a^{ij}{\mathrm{\partial }}_j{U}_m)=f_m-{\mathrm{\partial }}_i{G}_{i,m}$ together with $a^{33}\ge \alpha >0$ yields $a^{33}{\mathrm{\partial }}_{33}{U}_m={\mathrm{\partial }}_i{G}_{i,m}-f_m-{\sum }_{(i,j)\mathrm{\ne }(3,3)}{a}^{ij}{\mathrm{\partial }}_{ij}{U}_m-{\sum }_{i,j}({\mathrm{\partial }}_i{a}^{ij}){\mathrm{\partial }}_j{U}_m$ with every right-hand term in $L^2$, so all three second normal derivatives are $L^2$ and $U\in H^2$ locally; this does not presuppose a classical Robin derivative.

Bootstrap with boundary commutators: suppose $U\in H^{r+1}$ locally, $r\ge 1$; then $f\in H^r$ and $G\in H^{r+1}$ because the eigenvalue term is $\lambda JU$ and the coefficients are smooth. For a tangential multi-index $\alpha$ of order $r$ put $Z=D_y^{\alpha }U\in H^1$; its first two traces still vanish, and trace commutation is justified by extending localized $H^{r+1}$ classes with the integer-order half-space extension [F4], mollifying in that norm and passing the continuous $H^1$ trace identity to the limit. Applying the weak system of step 4.1 to $Z$ by tangential integration by parts leaves the same principal and Robin terms with bulk datum $D_y^{\alpha }f\in L^2$, divergence datum ${\tilde{G}}_i=D_y^{\alpha }{G}_i-{\sum }_{0<\gamma \le \alpha }\left(\genfrac{}{}{0px}{}{\alpha }{\gamma }\right)(D_y^{\gamma }{a}^{ij}){\mathrm{\partial }}_j{D}_y^{\alpha -\gamma }U\in H^1$ and boundary commutator $\tilde{h}=-{\sum }_{0<\gamma \le \alpha }\left(\genfrac{}{}{0px}{}{\alpha }{\gamma }\right)(D_y^{\gamma }\beta )T{D}_y^{\alpha -\gamma }{U}_3\in H^1(B)$: indeed $\mathrm{\mid }\alpha -\gamma \mathrm{\mid }\le r-1$, so the indicated derivative is $H^2$ and its trace and tangential derivative traces are $L^2$ by the $H^1$ trace and the just-proved commutation. Steps 5.1-6.1 then give $Z\in H^2$, controlling total order $r+2$ with at most two normal derivatives; differentiating the normal-recovery equation of step 6.1 and inducting on the number of normal derivatives at fixed total order (the non-$(3,3)$ terms have strictly fewer normal derivatives, and all differentiated coefficients and data have the available order) yields $U\in H^{r+2}$; nested patches and the finite boundary/interior cover give $u\in H^m(\mathrm{\Omega })$ for every integer $m$ with $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^m}\le C_{m,\mathrm{\Omega },\lambda }\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}$.

Classical conclusion: for each $j$ choose $m>j+3\mathrm{/}2$; the extension theorem and higher-order Sobolev embedding [F4] give continuous derivatives through order $j$ on the closure, compatible by weak-derivative uniqueness, and the fixed parameter-preserving smooth extension [F4] applied to the (time-constant) field yields one actual smooth ambient representative near the whole closure; therefore all jets have continuous one-sided boundary limits. The interior variational equation gives $\mathrm{curl}\mathrm{curl}u=\lambda u$ pointwise (The Laplacian of a $C^2$ function and of a $C^2$ vector field, [F3]), the tangential trace $\nu \times u=0$ is classical, and the boundary identity of step 1.1 tested with arbitrary smooth compactly supported normal traces, admissible by [F2], gives ${\mathrm{\partial }}_{\nu }{u}_n+\kappa u_n=0$ pointwise; the unit-normal collar extension ensures ${\mathrm{\partial }}_{\nu }{u}_n=({\mathrm{\partial }}_{\nu }u)\cdot \nu$, which agrees with the classical identity $\mathrm{div}u={\mathrm{\partial }}_{\nu }{u}_n+\kappa u_n$ for zero tangential trace, so $\mathrm{div}u=0$ holds classically.

The argument covers $\lambda =0$ and disconnected smooth domains, since no component is singled out and the local estimates are globalized over the finite cover; it makes no smoothness assertion for arbitrary finite-energy modal sums and no identification with a maximal tangential-trace realization, and the full Axiom of Choice is carried from the completion, trace, extension, Hilbert and Sobolev suppliers.