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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 !

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    $\begingroup$ I guess you mean $\partial A \,{\color{red}\cup}\, \partial B$. $\endgroup$ Jul 21, 2020 at 11:56
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    $\begingroup$ what definitely should work if you define $$N(f) = \inf_{G \in H^1(B^{\mathsf{c}}), \gamma_0 G = f} \|G\|_{H^1(B^{\mathsf{c}})}$$ $\endgroup$ Jul 21, 2020 at 12:26
  • $\begingroup$ Thanks @NathanaelSkrepek. Indeed, if I note your norm $$\tilde{N}(f)= \underset{G \in H^1(B^c), \gamma_0 G =f}{\inf} ||G||_{H^1(B^c)}$$ I am able to show that my norm $N$ is equivalent to $\tilde{N}$. However, I don't know how to prove that $\tilde{N}$ and $||.||_{H^{1/2}(\partial B)}$ are equivalent. $\endgroup$
    – Velobos
    Jul 22, 2020 at 8:24

2 Answers 2

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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 $\tilde{N}$.

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    $\begingroup$ Thanks for your answer. Am I allowed to use $||.||_*$ as a norm on $H^{1/2}(\partial B)$ ? I am aware that $$||u||^2_s=||u||^2_{L^2(\mathbb{R}^d)}+ \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{|u(x)-u(y)|^2}{||x-y||^{2(s+\frac{d}{2})}_{\mathbb{R}^d}} \ dx dy$$ defines a norm on the fractional sobolev space $H^s(\mathbb{R}^d)$ for $s \in ]0,1[$, but the demonstration I know uses the Fourier transform norm : $$\int_{\mathbb{R}^d} |\hat{u}(\xi)|^2(1+|\xi|^2)^s \ d\xi$$ but i'm not sure I can rely on the norm $||.||_*$ since Fourier transform should no more be possible ($\partial B$ is bounded). $\endgroup$
    – Velobos
    Jul 22, 2020 at 14:25
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    $\begingroup$ Also, I'm not sure but I think it should be better to use $$||f||_*=\left( ||f||_{L^2(\partial B)} + \int_{\partial B} \int_{\partial B} \frac{|f(x)-f(y)|^2}{||x-y||^d_{\mathbb{R}^d}} \ dx dy \right)^{\frac{1}{2}}$$. I actually found something that proves that $||.||_{*}$ is indeed a norm for $H^{1/2}(\partial B)$, but I don't know how to prove that it is equivalent to my norms $||.||_{H^{1/2}(\partial B)}$ and $||.||_{H^{1/2}(\partial B^c)}$. $\endgroup$
    – Velobos
    Jul 22, 2020 at 15:28
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    $\begingroup$ @Velobos I have seen it a lot that this norm is equivalent to the $H^{1/2}(\partial\Omega)$ that is given by the range norm of $H^1(\Omega)$, but I also don't know where you can find a reference for this. However, probably you will find references in arxiv.org/abs/1104.4345v3 I edited the norm in my post according to your suggestion $\endgroup$ Jul 23, 2020 at 8:19
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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 $\mathbb{R}^d$).

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