Let $\Phi \in H^2_\text{loc}(\mathbb{R}^3)$ and $\Omega$ a smooth bounded connected domain with surface $\Gamma:= \partial \Omega$.
We can define the following element $G$ in $H^{-{1/2}}(\Gamma)$ (the dual of $H^{{1/2}}(\Gamma)$) by :
$$G(w)=\int_{\Gamma} w \, \nabla \Phi \cdot n \ \mathrm{d}\gamma, \quad w \in H^{{1/2}}(\Gamma)$$ and $G$ is usually denoted $\frac{\partial \Phi}{\partial n}$.
Let $A$ be a bounded connected domain which countains $\bar{\Omega}$, I would like to prove that there exists a constant $C$ such that
$$\biggl\|\frac{\partial \Phi}{\partial n}\biggr\|_{H^{{-1/2}}(\Gamma)} = \sup_{w \in H^{{1/2}}(\Gamma)} \frac{|G(\omega)|}{\|w\|_{H^{{1/2}}(\Gamma)}} \leq C\|\nabla \Phi\|_{L^2(A)}.$$
If I give myself an uplift of $\omega$ (i.e a function $\psi \in H^1(\Omega)$ with $\gamma_0(\psi)=w$), I know I have access to a Green's formula through
$$G(w) = \int_{\Omega} \Delta \Phi \,\psi + \int_{\Omega} \nabla \Phi \cdot \nabla \psi. $$ Using this uplifting $\psi$, we have $||w||_{H^{{1/2}}(\Gamma)} \lesssim ||\psi||_{H^1(\Omega)}$. I also know, and I believe it will be the key to prove the statement, that for another uplift $\zeta \in H^1(A \setminus \Omega)$ such that $\gamma_1(\zeta)=w$ on $\Gamma$ and $\gamma_1 (\zeta) = 0$ on $\partial A$, we have
$$||w||_{H^{{1/2}}(\Gamma)} \lesssim \|\nabla \zeta\|_{H^1(A \setminus \Omega)}$$
The functions $\gamma_0, \gamma_1$ denotes the trace operators, respectively from $H^1(\Omega), \ H^1(A \setminus \Omega)$ to $H^{1/2}(\Gamma), H^{1/2}(\partial A \cup \partial \Omega)$.
Any help or remarks are welcomed !