The double-layer potential has the stated jumps and bounds

The affine class is nonempty: choose a bounded interior trace lift $L\varphi$ from [F1] and take $(u_0^{-},u_0^{+})=(-L\varphi ,0)$. Its trace jump is $-\varphi$ and its energy is at most $C\mathrm{\parallel }\varphi {\mathrm{\parallel }}_{H^{1\mathrm{/}2}}^2$. Let $Z={\mathcal{A}}_0$, the linear space of matching-trace pairs.

Equip $Z$ with $a_Z(v,w)={\int }_{{\mathrm{\Omega }}^{-}}\mathrm{\nabla }{v}^{-}\cdot \mathrm{\nabla }{w}^{-}+{\int }_{{\mathrm{\Omega }}^{+}}\mathrm{\nabla }{v}^{+}\cdot \mathrm{\nabla }{w}^{+}$. A matching pair glues to a function $v$ on ${\mathbb{R}}^3$. In each Lipschitz graph chart, integration by parts for smooth approximants gives the distributional derivative of the glued function as the two piecewise derivatives plus the interface term $({\gamma }^{+}{v}^{+}-{\gamma }^{-}{v}^{-})n_j$; trace continuity from [F1] passes this identity to $H^1$ functions, and the interface term vanishes. The glued function is in $L^6$: the exterior part is by definition in $L^6$, and the bounded interior part has a bounded $H^1({\mathbb{R}}^3)$ extension from [F1], to which [A1] applies after a bounded-support cutoff. Its global weak gradient is the two piecewise gradients. First show that any such $L^6$ function with global gradient in $L^2$ belongs to ${\dot{H}}^1$: take cutoffs ${\chi }_R=1$ on $B_R$, supported in $B_{2R}$, with $\mathrm{\mid }\mathrm{\nabla }{\chi }_R\mathrm{\mid }\le C\mathrm{/}R$. The gradient tail tends to zero, and Hölder gives $R^{-1}\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2(B_{2R}\setminus B_R)}\le C\mathrm{\parallel }v{\mathrm{\parallel }}_{L^6(B_{2R}\setminus B_R)}\to 0$. Thus ${\chi }_{R}v\to v$ in gradient norm; each compactly supported H1 cutoff is approximated in H1 by smooth compactly supported functions. Hence $v\in {\dot{H}}^1$, and [A1] gives $\mathrm{\parallel }v{\mathrm{\parallel }}_6\le C\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2$. A Cauchy sequence in $Z$ now glues to a Cauchy sequence in ${\dot{H}}^1$; its limit restricts to an interior H1 function and an exterior $E^{+}$ function, and bounded traces preserve the matching condition. Hence $Z$ is complete in $a_Z$. Also $a_Z(v,v)=0$ forces the glued $L^6$ function to be a constant and therefore zero, so $a_Z$ is a Hilbert inner product.

On $Z$, the functional $F(v)=-{\int }_{{\mathrm{\Omega }}^{-}}\mathrm{\nabla }{u}_0^{-}\cdot \mathrm{\nabla }{v}^{-}$ is bounded by $\mathrm{\parallel }\mathrm{\nabla }{u}_0^{-}{\mathrm{\parallel }}_2{a}_Z(v,v{)}^{1\mathrm{/}2}$. Apply [A2] to obtain a unique $z\in Z$ such that $a_Z(z,v)=F(v)$ for every $v\in Z$. Then $u=u_0+z\in {\mathcal{A}}_{\varphi }$ satisfies ${\int }_{{\mathrm{\Omega }}^{-}}\mathrm{\nabla }{u}^{-}\cdot \mathrm{\nabla }{v}^{-}+{\int }_{{\mathrm{\Omega }}^{+}}\mathrm{\nabla }{u}^{+}\cdot \mathrm{\nabla }{v}^{+}=0$ for every $v\in Z$. For any $v\in Z$, expansion gives $\mathcal{E}(u+v)=\mathcal{E}(u)+\mathcal{E}(v)$, so $u$ is the unique minimizer and $\mathcal{E}(u)\le \mathcal{E}(u_0)$. Taking compactly supported tests inside either side proves weak harmonicity.

The trace jump is the affine constraint. For any $g\in H^{1\mathrm{/}2}(\mathrm{\Gamma })$, restrict the bounded global lift $Lg$ of [F1] to both sides; the resulting matching pair lies in $Z$. Stationarity in step 3.1 and the weak conormal conventions [F4] give $⟨{\mathrm{\partial }}_n^{-}{u}^{-}-{\mathrm{\partial }}_n^{+}{u}^{+},g⟩=0$. Thus the conormals agree. The exterior energy trace bound [F2] and energy bound in step 3.1 bound ${\gamma }^{+}{u}^{+}$; the identity ${\gamma }^{-}{u}^{-}={\gamma }^{+}{u}^{+}-\varphi$ then bounds the interior trace and their average. For the common conormal use the interior lift alone: $\mathrm{\mid }⟨q_{\varphi },g⟩\mathrm{\mid }=\mathrm{\mid }{\int }_{{\mathrm{\Omega }}^{-}}\mathrm{\nabla }{u}^{-}\cdot \mathrm{\nabla }{L}^{-}g\mathrm{\mid }\le C\mathrm{\parallel }\mathrm{\nabla }{u}^{-}{\mathrm{\parallel }}_2\mathrm{\parallel }g{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}$, so $q_{\varphi }\in H^{-1\mathrm{/}2}$ with the asserted bound.

Step 3.1 supplies stationarity against every matching-trace finite-energy pair, in particular restrictions of every global smooth compactly supported function. The pair belongs to $H^1({\mathrm{\Omega }}^{-})\times E^{+}({\mathrm{\Omega }}^{+})$ and has trace jump $-\varphi$ by construction. Thus [F5] identifies it, with smooth representatives off $\mathrm{\Gamma }$, with the absolutely convergent double-layer kernel integral for every $\varphi \in H^{1\mathrm{/}2}(\mathrm{\Gamma })$. This includes facewise smooth densities reaching edges whenever they satisfy the global trace-space hypothesis.