Finite-energy transmission pairs have the double-layer kernel representation across polyhedral edges

By [F1], extension and a bounded-support cutoff put $u^{-}$ in $L^6({\mathrm{\Omega }}^{-})$; $u^{+}\in L^6({\mathrm{\Omega }}^{+})$ by definition. Thus $U\in L^6({\mathbb{R}}^3)$. Both pieces are $H^1$ in bounded boundary collars: on such a collar Holder gives local $L^2$ from $L^6$, and their gradients are already $L^2$. The fixed finite atlas also gives $\varphi \in L^2(\mathrm{\Gamma })\subset L^1(\mathrm{\Gamma })$.

For a compactly supported smooth $\eta$, integration by parts for one weak derivative gives $⟨{\mathrm{\partial }}_{j}U,\eta ⟩={\int }_{{\mathrm{\Omega }}^{-}}({\mathrm{\partial }}_j{u}^{-})\eta +{\int }_{{\mathrm{\Omega }}^{+}}({\mathrm{\partial }}_j{u}^{+})\eta +{\int }_{\mathrm{\Gamma }}\varphi n_j\eta dS\mathrm{.}$ Here is a polyhedral justification that introduces no hidden edge flux. In a graph chart integrate the vertical derivative by the one-dimensional fundamental theorem; for tangential derivatives use the graph change $x_3=t+\psi (x^{\mathrm{\prime }})$ and its piecewise affine chain rule. The graph boundary contribution is $(u^{+}-u^{-})n_{j}dS$. A chart partition sums these formulas; derivatives of the partition cancel. For the approximation, extend each collar restriction across the boundary using [F6], apply a bounded-support cutoff, and mollify. The derivative commutation and $L^2$ approximate-identity convergence in [F4], applied to the function and each weak first derivative, give $H^1$ convergence; the bounded trace in [F1] gives trace convergence. Thus the displayed identity passes from smooth restrictions to the given pieces. Internal planar-sector boundaries cancel because the restrictions of one $H^1$ function have the same interface trace. Only first derivatives are integrated and edges have surface measure zero; no derivative of the discontinuous face normal is taken. The imposed jump makes $u^{+}-u^{-}=\varphi$. Taking the divergence of the displayed distributional identity, and using stationarity for the smooth global test $\eta$, gives $⟨-\mathrm{\Delta }U,\eta ⟩={\int }_{\mathrm{\Gamma }}\varphi {\mathrm{\partial }}_n\eta dS\mathrm{.}$

Put $P(x)={\int }_{\mathrm{\Gamma }}{\mathrm{\partial }}_{n_y}{G}_3(x,y)\varphi (y)d{S}_y$ off $\mathrm{\Gamma }$. Since $\mathrm{\mid }{\mathrm{\partial }}_{n_y}{G}_3\mathrm{\mid }\le (4\pi {)}^{-1}\mathrm{\mid }x-y{\mathrm{\mid }}^{-2}$ and ${\int }_{B_R(a)}\mathrm{\mid }x-y{\mathrm{\mid }}^{-2}dx\le CR$ uniformly in $a,y$, [F3] defines $P\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^3)$ and gives ${\int }_{B_R(a)}\mathrm{\mid }P\mathrm{\mid }\le CR\mathrm{\parallel }\varphi {\mathrm{\parallel }}_{L^1(\mathrm{\Gamma })}\mathrm{.}$ For the uniform ball estimate, if $\mathrm{\mid }a-y\mathrm{\mid }\le 2R$ enclose the ball in $B_{3R}(y)$ and integrate radially; otherwise $\mathrm{\mid }x-y\mathrm{\mid }\ge \mathrm{\mid }a-y\mathrm{\mid }\mathrm{/}2>R\mathrm{/}2$, giving the same bound. Off $\mathrm{\Gamma }$ all kernel derivatives are uniformly bounded on compact sets, so differentiation under the surface integral proves smoothness and harmonicity. Differentiating [F2] with respect to the source variable gives ${\int }_{{\mathbb{R}}^3}{\mathrm{\partial }}_{n_y}{G}_3(x,y)(-\mathrm{\Delta }\eta (x))dx={\mathrm{\partial }}_n\eta (y)\mathrm{.}$ This differentiation is valid distributionally: $\mathrm{\mid }x-y{\mathrm{\mid }}^{-2}$ is locally volume integrable, and integration by parts outside a ball of radius $ϵ$ leaves a boundary term $O(ϵ)$ for the first derivative of $G_3$. Fubini using that same local integrability therefore yields $⟨-\mathrm{\Delta }P,\eta ⟩={\int }_{\mathrm{\Gamma }}\varphi {\mathrm{\partial }}_n\eta dS$.

Set $w=U-P$. Steps 2.1 and 2.2 show $\mathrm{\Delta }w=0$ distributionally on the whole space. Choose a nonnegative unit-mass smooth bump $\rho$ supported in $B_1$ and let $w_{ϵ}=w\ast {\rho }_{ϵ}$. By [F4], $w_{ϵ}$ is smooth and harmonic everywhere: $\mathrm{\Delta }{w}_{ϵ}(x)=⟨w,\mathrm{\Delta }{\rho }_{ϵ}(x-\cdot )⟩=0$. Holder and step 2.2 give, for fixed $ϵ$, ${\int }_{B_R(a)}\mathrm{\mid }{w}_{ϵ}\mathrm{\mid }\le C(R+ϵ{)}^{5\mathrm{/}2}\mathrm{\parallel }U{\mathrm{\parallel }}_6+C(R+ϵ)\mathrm{\parallel }\varphi {\mathrm{\parallel }}_1\mathrm{.}$ For a smooth harmonic $h$, differentiation of its spherical average gives $\frac{d}{dr}[(4\pi {)}^{-1}{\int }_{S^2}h(a+r\omega )d\omega ]=(4\pi r^2{)}^{-1}{\int }_{B_r(a)}\mathrm{\Delta }h=0$ by [F5]. Its limit at zero is $h(a)$, and radial integration gives the ball mean-value formula. Apply this to $h=w_{ϵ}$ to obtain $\mathrm{\mid }{w}_{ϵ}(a)\mathrm{\mid }\le \mathrm{\mid }{B}_R{\mathrm{\mid }}^{-1}{\int }_{B_R(a)}\mathrm{\mid }{w}_{ϵ}\mathrm{\mid }\to 0$ as $R\to \mathrm{\infty }$. Thus $w_{ϵ}=0$. Localizing $w$ to a slightly larger bounded set and applying the $L^1$ approximate-identity convergence of [F4] gives $w_{ϵ}\to w$ in $L^1$ on every compact set. Hence $U=P$ almost everywhere. The smooth harmonic $P$ supplies the asserted representatives.

Fix a face-interior point $x$ and select a planar smooth cutoff $\chi$ supported in that face and equal to one in a neighborhood of $x$. Split $\varphi =\chi \varphi +(1-\chi )\varphi$. The second density is supported a positive distance from $x$, so its potential extends smoothly through a neighborhood of $x$ and has boundary value its kernel integral. For the first density, choose planar coordinates with outward normal $e_3$. Direct differentiation gives the potential $\frac{1}{2}{P}_t\ast (\chi \varphi )$ for $t>0$ and $-\frac{1}{2}{P}_{\mathrm{\mid }t\mathrm{\mid }}\ast (\chi \varphi )$ for $t<0$, where $P_t(z)=t\mathrm{/}(2\pi (\mathrm{\mid }z{\mathrm{\mid }}^2+t^2{)}^{3\mathrm{/}2})$. Its mass is one by [F5], while its mass outside each fixed disk tends to zero by direct radial integration. Uniform continuity of the compactly supported $C^1$ density therefore gives the two limits $\pm \varphi (x)\mathrm{/}2$, locally uniformly in the face interior. The same-face boundary kernel is identically zero, and the complementary integral is nonsingular near $x$, proving the claimed $K_{\mathrm{k}\mathrm{e}\mathrm{r}}$ formula. Local uniform convergence also implies local $L^2$ convergence. On a compact tangential set $K$, parametrize either side by its nonnegative normal distance $t$. For $t>0$ the $H^1$ slice identity $\mathrm{\parallel }f(\cdot ,t)-\gamma f{\mathrm{\parallel }}_{L^2(K)}^2\le t{\int }_0^t\mathrm{\parallel }{\mathrm{\partial }}_{s}f(\cdot ,s){\mathrm{\parallel }}_{L^2(K)}^{2}ds$ follows from the fundamental theorem and the $H^1$ approximation constructed in step 2.1; hence those same local $L^2$ limits are the Sobolev traces. Exhaust each open face by compact subsets; the finite edge and vertex set has area zero. This proves the almost-everywhere trace-average assertion.

Finally, facewise $C^1$ functions agreeing on shared edges are continuous piecewise $C^1$ in each piecewise affine graph chart. Their weak first derivatives are the bounded sectorwise derivatives: integration by parts on the planar sectors cancels all shared-edge boundary terms. After the chart cutoffs they belong to $H^1$ on ${\mathbb{R}}^2$, hence to its $H^{1\mathrm{/}2}$ Bessel space by the Fourier weight inequality. Thus such densities satisfy the stated $H^{1\mathrm{/}2}$ hypothesis even when nonzero at edges and vertices. No face-interior-core density assertion at the critical exponent is needed.