Estimate of the norm of the gradient in $H^1(\Omega)^n$ in terms of the norm of the function in $H^2(\Omega)$

Question

Let $\Omega \subset \mathbb{R}^n$ be bounded and $u \in H^2(\Omega).$

Does it hold that $$ \Vert \nabla u \Vert_{H^1(\Omega)^n}^2 \leq \Vert u \Vert_{H^2(\Omega)}^2? $$

In my attempt, I always find a constant in the right-hand side, meaning $$ \Vert \nabla u \Vert_{H^1(\Omega)^n}^2 \leq C \Vert u \Vert_{H^2(\Omega)}^2 $$ where $C>0$.