Weak Neumann and transmission existence with interface charges

Boundedness of $\mathrm{\ell }$: $\mathrm{\mid }\mathrm{\ell }(v)\mathrm{\mid }\le \mathrm{\parallel }\rho {\mathrm{\parallel }}_2\mathrm{\parallel }v{\mathrm{\parallel }}_2+\mathrm{\parallel }g{\mathrm{\parallel }}_{H^{-1\mathrm{/}2}}\mathrm{\parallel }Tv{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}\le C(1+\mathrm{\parallel }\rho {\mathrm{\parallel }}_2+\mathrm{\parallel }g{\mathrm{\parallel }}_{H^{-1\mathrm{/}2}})\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}$ by the trace bound [F1] and Cauchy-Schwarz [F4].

Necessity of compatibility: the constant function $1$ lies in $H^1(\mathrm{\Omega })$ with $\mathrm{\nabla }1=0$, so testing the equation with $v=1$ gives $0=\mathrm{\ell }(1)$; hence $\mathrm{\ell }(1)=0$ is necessary for existence.

Assume now the compatibility condition $\mathrm{\ell }(1)=0$. Let $H_{\ast }^1=\{v\in H^1(\mathrm{\Omega }):{\int }_{\mathrm{\Omega }}v=0\}$, a closed subspace, and restrict the symmetric form $a(w,v)={\int }_{\mathrm{\Omega }}\mathrm{\nabla }v\cdot \epsilon \mathrm{\nabla }w$ and the functional $\mathrm{\ell }$ to it; $a$ is bounded by $\beta$ in gradient norm: the nonnegative quadratic $b_x(\xi +t\eta ,\xi +t\eta )$ for $b_x(\xi ,\eta )=\xi \cdot \epsilon (x)\eta$ gives $\mathrm{\mid }{b}_x(\xi ,\eta )\mathrm{\mid }\le \beta \mathrm{\mid }\xi \mathrm{\mid }\mathrm{\mid }\eta \mathrm{\mid }$, and Cauchy-Schwarz [F4] gives $\mathrm{\mid }a(w,v)\mathrm{\mid }\le \beta \mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_2\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2$. It is coercive with $a(v,v)\ge \alpha \mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2^2\ge c\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}^2$ by the equivalence of norms of [F2], and $\mathrm{\ell }$ remains bounded by step 1.1, so the solver [F3] gives a unique $u_{\ast }\in H_{\ast }^1$ with $a(u_{\ast },v)=\mathrm{\ell }(v)$ for every $v\in H_{\ast }^1$.

Extension to all of $H^1$: every $v\in H^1$ decomposes as $v=v_{\ast }+c$ with $v_{\ast }\in H_{\ast }^1$ and constant $c=v_{\mathrm{\Omega }}$; then $a(u_{\ast },v)=a(u_{\ast },v_{\ast })+a(u_{\ast },c)=a(u_{\ast },v_{\ast })=\mathrm{\ell }(v_{\ast })=\mathrm{\ell }(v_{\ast })+c\mathrm{\ell }(1)=\mathrm{\ell }(v)$ by the compatibility assumed in step 2.2 and $\mathrm{\nabla }c=0$, so $u_{\ast }$ solves the full weak Neumann problem; normalising $\int u_{\ast }=0$ selects this representative.

Uniqueness: for two solutions $u_1,u_2$ the difference $w$ satisfies $\int \mathrm{\nabla }v\cdot \epsilon \mathrm{\nabla }w=0$ for all $v\in H^1$; testing $v=w$ gives $\alpha \mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_2^2\le \int \mathrm{\nabla }w\cdot \epsilon \mathrm{\nabla }w=0$, so $\mathrm{\nabla }w=0$ and, by the mean-zero inequality [F2], $w-w_{\mathrm{\Omega }}=0$, i.e. $w$ is constant; hence solutions are unique up to additive constants and the mean-zero normalisation makes the solution unique.

Finitely many connected components: on a bounded $C^1$ domain with components ${\mathrm{\Omega }}_1,\dots ,{\mathrm{\Omega }}_N$, $H^1(\mathrm{\Omega })$ is the direct sum of the $H^1({\mathrm{\Omega }}_j)$ extended by zero, the gradient form and the functional $\mathrm{\ell }$ split, and the trace space splits accordingly, so the argument of steps 1.1-4.1 applies on each component separately: compatibility $\mathrm{\ell }({\mathbf{1}}_{{\mathrm{\Omega }}_j})=0$ is required on each component and the solution is unique up to one constant per component, fixed by the per-component mean-zero normalisation.

Transmission: include the bounded interface functional $⟨\sigma ,T_{\mathrm{\Sigma }}v⟩$ in $\mathrm{\ell }$. For Dirichlet data use the right-inverse lift $G$ of [F1] and solve $a(w,v)=\mathrm{\ell }(v)-a(G,v)$ on $H_0^1$ by [F3] and zero-boundary coercivity of [F2]; then $u=G+w$ has the prescribed outer trace and solves the full interface weak problem. For Neumann data apply steps 2.2–5.1 with this modified $\mathrm{\ell }$ and its componentwise compatibility. The one-sided traces coincide: localized global $H^1$ functions can be approximated by smooth ambient restrictions using [F4]; their traces coincide on the interface, and bounded one-sided trace maps pass that equality to the $H^1$ limit.

Classical jump: where the transmission solution is classically smooth on each side of an interface $\mathrm{\Sigma }$ with unit normal $n$ pointing from the side labelled minus to the side labelled plus, and $[W]=W_{+}-W_{-}$, apply the divergence theorem to $\mathrm{\nabla }v\cdot \epsilon \mathrm{\nabla }u$ on each side and subtract the weak equation; the bulk terms cancel and the remaining boundary identity is $n\cdot [\epsilon \mathrm{\nabla }u]=-\sigma$, i.e. $n\cdot [D]=\sigma$ for the explicitly defined mathematical flux $D:=-\epsilon \mathrm{\nabla }u$, since $[D]=-[\epsilon \mathrm{\nabla }u]$; this is conditional on the one-sided classical smoothness, and general piecewise classical regularity remains a separate requirement.