This question [has been posted on Mathematics Stack Exchange](https://math.stackexchange.com/q/4529834/228959) but got no response, and so I put it here.

When I read the paper "On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation" by Igor Chueshov, Matthias Eller and Irena Lasiecka, I encounter a difficulty:

Suppose that $\Omega$ is a simple connected domain with smooth boundary, the authors introduced a Robin Laplacian operator $$\Delta_{R}\colon L^{2}(\Omega)\to L^{2}(\Omega).$$This is an unbounded operator with the domain $$D(\Delta_{R})=\bigl\{u\in H^{2}(\Omega)\colon \partial_{\nu}u+u=0\ on\ \partial\Omega\bigr\}.$$ Moreover, the Robin Laplacian can be extended to a continuous operator $\Delta_{R}\colon H^{1}(\Omega)\to H^{1}(\Omega)'$ by $$(-\Delta_{R}u,v)_{L^{2}(\Omega)}=(\nabla u,\nabla v)_{L^{2}(\Omega)}+\langle u,v\rangle_{L^{2}(\partial\Omega)}.$$ Then the authors say "this extension is the duality map $H^{1}(\Omega)$ into $(H^{1}(\Omega))'$" when we equip $H^{1}(\Omega)$ the norm $$\|u\|^* =\sqrt{(\nabla u,\nabla u)_{L^{2}(\Omega)}+\langle u,u\rangle_{L^{2}(\partial\Omega)}}.$$
So, firstly, I want to know if this norm is equivalent to the usual Sobolev norm
$$\|u\|=\sqrt{(\nabla u,\nabla u)_{L^{2}(\Omega)}+(u,u)_{L^{2}(\Omega)}}$$ when $u$ satisfies the Robin boundary condition? Next, the authors said $$D((-\Delta_{R})^{\frac{1}{2}})\sim H^{1}(\Omega),$$ why? The authors offered a reference but it is French, but I don't understand French. The reference is 
"Grisvard, P.: Characterisation de Quelques Espaces d'interpolation. Archives Rational Mechanics and Analysis (1967) #26, pp.40-63"
 
Any comments and hints are welcome, thank you very much!