The hypersingular operator is elliptic modulo constants

If $u=D\varphi$ and $v=D\psi$, their sum belongs to ${\mathcal{A}}_{\varphi +\psi }$ and satisfies the stationarity equation because each summand does; uniqueness of the minimizer gives $D(\varphi +\psi )=D\varphi +D\psi$, and scalar homogeneity follows in the same way. The interior weak Green formula and the exterior finite-energy Green identity justified in the weak layer definition give ${\int }_{{\mathrm{\Omega }}^{-}}\mathrm{\nabla }{u}^{-}\cdot \mathrm{\nabla }{v}^{-}+{\int }_{{\mathrm{\Omega }}^{+}}\mathrm{\nabla }{u}^{+}\cdot \mathrm{\nabla }{v}^{+}=⟨{\mathrm{\partial }}_n^{-}u,{\gamma }^{-}v⟩-⟨{\mathrm{\partial }}_n^{+}u,{\gamma }^{+}v⟩\mathrm{.}$ Since $u$ has common conormal $q_{\varphi }$ and ${\gamma }^{-}v-{\gamma }^{+}v=-\psi$, the right side is $-⟨q_{\varphi },\psi ⟩=⟨W\varphi ,\psi ⟩$. The left side is symmetric in $u,v$, so $W$ is symmetric. Setting $\psi =\varphi$ gives $⟨W\varphi ,\varphi ⟩=\mathcal{E}(D\varphi )\ge 0$.

Let $u=D\varphi$ and set $a=u_{{\mathrm{\Omega }}^{-}}^{-}$. By [F3], $\mathrm{\parallel }{\gamma }^{-}{u}^{-}-a{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}\le C\mathrm{\parallel }\mathrm{\nabla }{u}^{-}{\mathrm{\parallel }}_2$. By [F4], $\mathrm{\parallel }{\gamma }^{+}{u}^{+}{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}\le C\mathrm{\parallel }\mathrm{\nabla }{u}^{+}{\mathrm{\parallel }}_2$. The jump condition gives $\varphi +a=-({\gamma }^{-}{u}^{-}-a)+{\gamma }^{+}{u}^{+}$, so ${\inf }_{c\in \mathbb{R}}\mathrm{\parallel }\varphi -c{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}\le \mathrm{\parallel }\varphi +a{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}\le C\mathcal{E}(D\varphi {)}^{1\mathrm{/}2}\mathrm{.}$ Together with step 1.1 this proves the quotient ellipticity inequality. If $\varphi$ is constant, the pair $(-\varphi ,0)$ belongs to ${\mathcal{A}}_{\varphi }$ and has zero energy, so it is the unique minimizer and step 1.1 gives $W\varphi =0$. Conversely, if $W\varphi =0$, the ellipticity inequality makes $\varphi$ a limit in $H^{1\mathrm{/}2}$ of constants; the constant subspace is closed because it is finite dimensional, so $\varphi$ is constant.

Put $\bar{\varphi }=\mathrm{\mid }\mathrm{\Gamma }{\mathrm{\mid }}^{-1}{m}_{\mathrm{\Gamma }}(\varphi )$ and ${\varphi }_0=\varphi -\bar{\varphi }$. The quotient norm controls $\mathrm{\parallel }{\varphi }_0{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}$: for any constant $a$, the $L^2$ bound gives $\mathrm{\mid }\bar{\varphi }-a\mathrm{\mid }\le \mathrm{\mid }\mathrm{\Gamma }{\mathrm{\mid }}^{-1\mathrm{/}2}\mathrm{\parallel }\varphi -a{\mathrm{\parallel }}_{L^2}$, hence $\mathrm{\parallel }{\varphi }_0{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}\le C\mathrm{\parallel }\varphi -a{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}$, and take the infimum. Also $\mathrm{\mid }\bar{\varphi }\mathrm{\mid }=\mathrm{\mid }{m}_{\mathrm{\Gamma }}(\varphi )\mathrm{\mid }\mathrm{/}\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }$. Thus $\mathrm{\parallel }\varphi {\mathrm{\parallel }}_{H^{1\mathrm{/}2}}^2\le C({\inf }_a\mathrm{\parallel }\varphi -a{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}^2+\mathrm{\mid }{m}_{\mathrm{\Gamma }}(\varphi ){\mathrm{\mid }}^2)$. By [F2], $W$ is bounded; the rank-one term is bounded because $\mathrm{\mid }{m}_{\mathrm{\Gamma }}(\varphi )\mathrm{\mid }\le \mathrm{\mid }\mathrm{\Gamma }{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }\varphi {\mathrm{\parallel }}_{L^2}\le C\mathrm{\parallel }\varphi {\mathrm{\parallel }}_{H^{1\mathrm{/}2}}$, so $b$ is bounded. Its coercivity follows from step 2.1 and the last decomposition.

The space $H^{1\mathrm{/}2}(\mathrm{\Gamma })$ is a real Hilbert space under the square-sum chart norm fixed in Lipschitz polyhedral boundary, orientation, and trace spaces. Apply [F5] to the bounded coercive form $b$ and the bounded functional $\psi \mapsto ⟨f,\psi ⟩$; it yields a unique solution for every $f\in H^{-1\mathrm{/}2}(\mathrm{\Gamma })$.