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Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with a $C^2-$ boundary $\partial \Omega= \Gamma$. For $f \in H^{1/2}(\Gamma)$, let $F \in H^1(\Omega)$ denote the weak solution of the Dirichlet problem:

$\begin{cases} \Delta F=0 & \text{in } \Omega\\ F=f & \text{on } \Gamma \\ \end{cases}$

and let $Nf:=\frac{\partial F}{\partial \eta}$ be the Neumann data on $\Gamma$. This defines a continuous map $N: H^{1/2}(\Gamma) \to H_{*}^{-1/2}(\Gamma) \equiv \{g \in H^{-1/2}(\Gamma) \vert \int_{\Gamma} g=0\}$

I have trouble understanding why $Nf \in H_{*}^{-1/2}(\Gamma)$. What I can see is that, $F \in H^1(\Omega)$ implies $F \in H^{1/2}(\Gamma)$ and moreover $\int_{\Gamma} Nf=\int_{\Gamma} \frac{\partial F}{\partial \eta}=\int_{\Omega} \Delta F =0$. But I don't understand why $\frac{\partial F}{\partial \eta} \in H^{-1/2}(\Gamma)$

EDIT: An idea just popped up inside my head, but I'd like if someone could confirm if it's correct. Applying Green's formula we find:

$\int_{\Omega} {\vert \nabla F \vert }^2= \int_{\Gamma} F\frac{\partial F}{\partial \eta}=\int_{\Gamma} f Nf $ for every $f\in H^{1/2}(\Gamma)$. Now since $\nabla F \in L^2(\Omega)$ we deduce that $Nf \in H^{-1/2}(\Gamma)$ by duality.

I 'm not familiar to this operator so I apologize in advance if I missed something from the definition. Any help or hint is much appreciated.

Thanks!

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$N$ is a first order pseudo-differential operator. Hence it maps Sobolev space $H^s(\Gamma)$ into $H^{s-1}(\Gamma)$. Furthermore, $N$ is elliptic, its principal symbol is the square root of the principal symbol of the Laplacian. A reference is Appendix C in PDE II of Michael E. Taylor.

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