Lipschitz finite-energy transmission pairs have the double-layer kernel representation

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 $\eta$, apply the Lipschitz first-order Green identity [F5] to each piece, localized by a finite smooth chart partition. The graph normals have opposite signs and are intrinsic on overlaps; partition derivative terms cancel. Thus $⟨{\mathrm{\partial }}_{j}U,\eta ⟩={\int }_{{\mathrm{\Omega }}^{-}}({\mathrm{\partial }}_j{u}^{-})\eta +{\int }_{{\mathrm{\Omega }}^{+}}({\mathrm{\partial }}_j{u}^{+})\eta +{\int }_{\mathrm{\Gamma }}({\gamma }^{+}{u}^{+}-{\gamma }^{-}{u}^{-})n_j\eta dS$. Both pieces are $H^1$ up to the boundary by step 1.1, so the approximation and bounded endpoint trace in [F5] apply. The imposed jump gives ${\gamma }^{+}{u}^{+}-{\gamma }^{-}{u}^{-}=\varphi$. Taking the divergence distributionally and using the global smooth-test stationarity therefore gives $⟨-\mathrm{\Delta }U,\eta ⟩={\int }_{\mathrm{\Gamma }}\varphi {\mathrm{\partial }}_n\eta dS$. No planar face decomposition, surface-normal differentiation or pointwise jump formula is used.

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.