Different norms for the $H^{1/2}$ sobolev spaces

Question

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.

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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 !

Answer 1

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 $\mathbb{R}^d$).

Answer 2

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 $\tilde{N}$.