I am a little bit confused about different definitions of the trace space $H^{3/2}(\partial \Omega)$, and I hope I can find some simple examples on how to explicitly compute these norms for simple geometries.
For example, let $u:\Omega\to \mathbb{R}$ be a $C^1$ function on a sufficiently smooth domain $\Omega$, say $\Omega=B_R(0)\subset \mathbb{R}^2$, such that the traces of its value and its derivatives equal to themselves. May I know how to explicitly compute the norm $\|u\|_{H^{3/2}(\partial \Omega)}$ on the circle $\partial \Omega=\{(x,y)\mid x^2+y^2=R^2\}$? I know for sufficiently smooth $u$, the norm can be defined as $$ \|u\|^2_{H^{3/2}(\partial \Omega)}=\int_{\partial\Omega} |u|^2\,dS+\int_{\partial \Omega} \sum_{i=1}^2 |\partial_i u|^2\,dS+ \sum_{i=1}^2 \int_{\partial \Omega}\int_{\partial \Omega}\frac{|\partial_i u(x)-\partial_i u(y)|^2}{|x-y|^2}\,dS_xdS_y, $$ where $dS_x$ is the boundary measure. But what do these $\partial_iu$ mean? Is it simply the derivatives of $u$ on $\Omega$ restricted on $\partial \Omega$? I thought the definition of $\partial_i$ must involve the geometry of the domain.