Assume $\Omega\subset R^n $ is a nice domain in $R^n$. I know that we can have the following control in sobolev space:
$ \|u\|_{L^2(\Omega)} \leq C \|\nabla u \|_{L^2(\Omega)}, \forall u \in H^1_0(\Omega) \text{ or } u \in H^1(\Omega) \bigcap \{u: \int_\Omega u dx = 0 \} $
My question is if we can prove the following: for any $\epsilon > 0$, there exists $C > 0$ such that $ \|u\|_{L^2(\Omega)} \leq \epsilon \|u\|_{L^2(\partial \Omega)} + C \|\nabla u\|_{L^2(\Omega)}, \forall u \in H^1(\Omega)$
This is not true: Assume that the claim holds for some $\epsilon$. Now set $u\equiv 1$. Then $$ |\Omega|^{1/2} \le \epsilon |\partial \Omega|^{1/2}, $$ where $|\Omega|$ and $|\partial \Omega|$ are the $d$ and $d-1$ dimensional measures of these sets, respectively.
This gives a lower bound on $\epsilon$.