Suppose that $\Omega\subset\mathbb{R}^d$ is a smooth, bounded, and connected domain. Let \begin{equation} H=\{u\in H^2(\Omega):\int_\Omega u(x) dx=0 ~\text{and}~ \nabla u\cdot v=0~ \text{on}~\partial\Omega\}. \end{equation} Show that $H$ is a Hilbert space, and prove that there exists $C>0$ such that for any $u\in H$, \begin{equation} ||u||_{H^1(\Omega)}\le C\sum_{|\alpha|=2} ||D^\alpha u||_{L^2(\Omega)}. \end{equation}
I can prove the space is a Hilbert space. How do I put my hands on the inequality?
Suppose that the inequality doesn't hold, so for all $n\in\mathbb{N}$ there exists a $u_n\in H$ such that $$ \|u_n\|_{H^1}>n\sum_{|\alpha|=2}\|D^\alpha u_n\|_{L^2}. $$ Further, normalizing, we can assume that $\|u_n\|_{H^1}=1$ for all $n\in\mathbb{N}$. So, $$ \frac{1}{n}>\sum_{|\alpha|=2}\|D^\alpha u_n\|_{L^2} $$ for all $n\in\mathbb{N}$. You should be able to derive a contradiction from here by taking a convergent subsequence of $(u_n)$ in $L^2$ and using the fact that $\int_\Omega u(x)dx=0$ for all $u\in H$.