Different norms for the $H^{1/2}$ sobolev spaces I am looking for references for the following results, which i believe to be true :
Let $B$ a Lipschitz domain in $\mathbb{R}^d$, $f \in H^{1/2}(\partial B)$. We note $\gamma_0 : H^1(B) \mapsto H^{1/2}(\partial B)$ the natural trace application for $B$. I know we can provide $H^{1/2}(\partial B)$  with the following norm :
\begin{equation}
    ||f||_{H^{1/2}(\partial B)} = \underset{G \in H^1(B) \atop
     \gamma_0(G)|_{\partial B}=f}{\inf}||G||_{H^1(B)}.
\end{equation}
Let $A$ a bounded open set with regular boundaries such as $B \subset A$. We note $N : H^{1/2}(\partial B) \mapsto \mathbb{R}$ defined by :
\begin{equation}
    N(f) = \underset{G \in H^1(A \setminus B) \atop
     \tilde{\gamma_0}(G)|_{\partial B}=f \text{ et } \tilde{\gamma_0}(G)|_{\partial A}=0}{\inf}||\nabla G||_{(L^2(A \setminus B))^{d^2}}.
\end{equation}
where $\tilde{\gamma_0} : H^1(A \setminus B) \mapsto H^{1/2}(\partial A \cup \partial B)$ is the natural trace application for the space $A \setminus B$.
I am looking to prove that $N$ is a norm for $H^{1/2}(\partial B)$ and that $N$ and $||.||_{H^{1/2}(\partial B)}$ are equivalent norms.

In concrete terms, this results means that it is the same to define a norm on $H^{1/2}(\partial B)$ weither you extend $f$ in the exterior ($A \setminus B)$ or interior ($B$) of $\partial B$.
I already looked in, among other sources :
Galdi, Giovanni P., An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I: Linearized steady problems, Springer Tracts in Natural Philosophy. 38. New York, NY: Springer-Verlag. xi, 450 p. (1994). ZBL0949.35004.
Evans, Lawrence C., Partial differential equations, Graduate Studies in Mathematics. 19. Providence, RI: American Mathematical Society (AMS). xvii, 662 p. (1998). ZBL0902.35002.
Any help or information are welcomed !
 A: Given that $N$ and $\tilde{N}$ are equivalent as you mentioned in the comments, I think this proves your statement:
The norm
\begin{equation}
    \|f\|_{H^{1/2}(\partial B)} := \inf_{G \in H^1(B),\gamma_0(G)=f}\|G\|_{H^1(B)}.
\end{equation}
is equivalent to
\begin{align*}
\|f\|_{\ast} := \left(\|f\|^2_{L^2(\partial B)} + \int_{\partial B} \int_{\partial B} \frac{|f(x)-f(y)|^2}{\|x-y\|_{\mathbb{R}^n}^n} 
\,\mathrm{d}x\,\mathrm{d}y\right)^{\frac{1}{2}}
\end{align*}
which is in turn equivalent to
\begin{align*}
  \|f\|_{H^{1/2}(\partial (B^{\mathsf{c}}))}
  := \inf_{G \in H^1(B^c),\gamma_0(G)=f}\|G\|_{H^1(B^{\mathsf{c}})}.
\end{align*}
which is $\tilde{N}$.
A: There is maybe an alternative way : showing that for $g \in H^1(\mathbb{R}^d)$, the trace on $\partial B$ is the same for $g$ which is seen as an element of $H^1(B)$ or as an element of $H^1(B^c)$. If you denote those two elements $\gamma g$ and $\widetilde{\gamma} g$, you can play with infimum and the continuous extension operators (from $B$ to $\mathbb{R}^d$ and $B^c$ to $\mathbb{R}^d$) to show that you have indeed
$$||\cdot||_{H^{1/2}(\partial B)} \sim \widetilde{N}$$
(with your previous notations).
The point is now to see that the traces above are the same. I think the main point is to understand why the trace actually only depends on $\partial B$ as a manifold and not on $B$ or $B^c$. I suggest you have a look to the following famous book :
F. Boyer, P. Fabrie : Mathematical tools for the study of the incompressible Navier-Stokes equations and related models, Applied Mathematical Sciences, vol. 183, Springer (2013)
and more precisely Section 2.5.1. (even if it relies on some previous notations inside the book...) I think that you can convince yourself that the construction of the trace yields the same map for a $g \in H^1(\mathbb{R}^d)$ (the partition of unity you use is the same and you control all the norms on $B$ or $B^c$ by the same norms on $\mathbb{R}^d$).
