I know two theorems about the trace inequality.
Suppose that $\Omega$ is a bounded domain with smooth boundary.
One is that: $$ \gamma_0(H^1(\Omega)) = H^{\frac{1}{2}}(\partial \Omega) $$ where $\ \ \ \ \gamma_0:H^1(\Omega) \rightarrow L^2(\partial \Omega) \quad\quad$ $ u\rightarrow u|_{\partial \Omega} $.
The other is that: $$ \gamma_1(H^2(\Omega)) = H^{\frac{1}{2}}(\partial \Omega) $$ where $\ \ \ \gamma_1:H^2(\Omega) \rightarrow L^2(\partial \Omega) \quad\quad\quad$ $ u\rightarrow \frac{\partial u}{\partial n}|_{\partial \Omega}. $
So if given any $v\in H^{\frac{1}{2}}(\partial \Omega)$, I know that there exist some $\{\phi_i\}\in H^2(\Omega)$ such that $$ \frac{\partial \phi}{\partial n}|_{\partial \Omega} = v $$ and $\|\phi_i\|\leq\|v\|_{L^2(\Omega)}$.
Question: Dose exist some $\phi\in H_0^1$ in $\{\phi_i\}$? Intuitively it's right because I think the normal derivative won't affect the boundary value. I really hope for some reference and the answer or some motivation. Thanks a lot.