Assume $\Omega \subset \mathbb{R}^{3}$ is an open bounded smooth domain. I know that the following function defines a norm on $H^{2}(\Omega)$:
$$\| \cdot \|^{2}_{H^{2}} := \| \cdot \|^{2} + \| \Delta \cdot \|^{2} $$ where $\| \cdot \|$ denotes the $L^{2}$ norm.
Is it true that the following defines an equivalent norm on $H^{4}(\Omega)$?
$$\| \cdot \|^{2}_{H^{4}} := \| \cdot \|^{2}_{H^{2}} + \| \Delta \cdot \|^{2}_{H^{2}} $$