# 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 :

$$$$||f||_{H^{1/2}(\partial B)} = \underset{G \in H^1(B) \atop \gamma_0(G)|_{\partial B}=f}{\inf}||G||_{H^1(B)}.$$$$ Let $$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 :

$$$$N(f) = \underset{G \in H^1(A \setminus B) \atop \tilde{\gamma_0}(G)|_{\partial B}=f \text{ et } \tilde{\gamma_0}(G)|_{\partial A}=0}{\inf}||\nabla G||_{(L^2(A \setminus B))^{d^2}}.$$$$ where $$\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 !

• I guess you mean $\partial A \,{\color{red}\cup}\, \partial B$. Jul 21, 2020 at 11:56
• what definitely should work if you define $$N(f) = \inf_{G \in H^1(B^{\mathsf{c}}), \gamma_0 G = f} \|G\|_{H^1(B^{\mathsf{c}})}$$ Jul 21, 2020 at 12:26
• 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. Jul 22, 2020 at 8:24

Given that $$N$$ and $$\tilde{N}$$ are equivalent as you mentioned in the comments, I think this proves your statement: The norm $$$$\|f\|_{H^{1/2}(\partial B)} := \inf_{G \in H^1(B),\gamma_0(G)=f}\|G\|_{H^1(B)}.$$$$ is equivalent to \begin{align*} \|f\|_{\ast} := \left(\|f\|^2_{L^2(\partial B)} + \int_{\partial B} \int_{\partial B} \frac{|f(x)-f(y)|^2}{\|x-y\|_{\mathbb{R}^n}^n} \,\mathrm{d}x\,\mathrm{d}y\right)^{\frac{1}{2}} \end{align*} which is in turn equivalent to \begin{align*} \|f\|_{H^{1/2}(\partial (B^{\mathsf{c}}))} := \inf_{G \in H^1(B^c),\gamma_0(G)=f}\|G\|_{H^1(B^{\mathsf{c}})}. \end{align*} which is $$\tilde{N}$$.

• 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). Jul 22, 2020 at 14:25
• 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 $||.||_{H^{1/2}(\partial B^c)}$. Jul 22, 2020 at 15:28
• @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 Jul 23, 2020 at 8:19

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 $$\mathbb{R}^d$$).