This question has been posted on Mathematics Stack Exchange 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 $$\Delta_{R}:L^{2}(\Omega)\to L^{2}(\Omega).$$This is an unbounded operator with the domain $$D(\Delta_{R})=\{u\in H^{2}(\Omega):\partial_{\nu}u+u=0\ on\ \partial\Omega\}.$$Moreover, the Robin Laplacian can be extended to a continuous operator $\Delta_{R}:H^{1}(\Omega)\to H^{1}(\Omega)'$ by $$(-\Delta_{R}u,v)_{L^{2}(\Omega)}=(\nabla u,\nabla v)_{L^{2}(\Omega)}+<u,v>_{L^{2}(\partial\Omega)}.$$ 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 $$\|u\|=\sqrt{(\nabla u,\nabla v)_{L^{2}(\Omega)}+<u,v>_{L^{2}(\partial\Omega)}}.$$ So, firstly, I want to know if this norm is equivalent to the usual Sobolev norm $$\|u\|=\sqrt{(\nabla u,\nabla v)_{L^{2}(\Omega)}+(u,v)_{L^{2}(\Omega)}}$$ 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 Esoaces d'interpolation. Archives Rational Mechanics and Analysis 1967,26,40-63"

Any comments and hints are welcome, thank you very much!!!