I am trying to prove an equivalence between two norms in the Sobolev space $H^1(\Omega)$ over a bounded Lipschitz domain $\Omega$, namely the standard norm $||u||_{H^1(\Omega)}^2=\int_{\Omega} u^2 \,dx + \int_{\Omega} |\nabla u|^2\,dx$ and the norm $||u||_{\partial}^2= \int_{\partial \Omega} u_{|\partial \Omega}^2 \,d\sigma + \int_{\Omega} |\nabla u|^2\,dx$
The second sumands are the same in both cases, so since in the first one we integrate the positve function $u^2$ over a larger domain, it is clear that $||u||_{\partial} \leq ||u||_{H^1(\Omega)}$, so it now suffices to find a positive constant $C$ such that $C||u||_{H^1(\Omega)}\leq ||u||_{\partial}$.
Now we can use the following projection theorem: given a Hilbert space $H$ and a closed subspace $V\subset H$, for every $X\in H$ there exists a unique $P_Vx\in V$ such that $||x-P_vx||=\inf_{v\in V} \{||v-x||\}$ and besides $x-P_Vx\in V^{\perp}$.
In our case, this yields a decomposition $u=u_0+E(u_{|\partial \Omega})$ where $E(u_{|\partial \Omega})$ is an extension to $\Omega$ of the restriction $u_{|\partial \Omega}$ and $u_0\in H_0^1(\Omega)$ and therefore $||u||_{H^1(\Omega)}^2=||u_0||_{H^1(\Omega)}^2+||E(u_{|\partial \Omega})||_{H^1(\Omega)}^2$
Note that by the projection theorem we have that $||E(u_{|\partial \Omega})||_{H^1(\Omega)}^2=\inf\{||v||_{H^1(\Omega)}: v\in H^1(\Omega), v_{|\partial \Omega}=u_{|\partial \Omega}\} \stackrel{def}{=} ||u_{|\partial \Omega}||_{H^{1/2}(\partial \Omega)}$ Does this lead towards our purpose?
Thanks in advance for any insight.