# Question on definition of Dirichlet to Neumann operator

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!

$$N$$ 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.