Equivalence of norms in the space $H_\Delta(\Omega)$

Question

Let $\Omega \subset \mathbb{R}^n$ is an open bounded domain with smooth boundary $\Gamma$. Consider the following space $$H_\Delta(\Omega)=\{u\in L^2(\Omega) : \Delta u \in L^2(\Omega)\},$$ with the norm $$\|u\|_{H_\Delta(\Omega)}^2 := \|u\|_{L^2(\Omega)}^2+\|\Delta u\|_{L^2(\Omega)}^2,$$

Some authors define the previous space as $$H^1_\Delta(\Omega)=\{u\in H^1(\Omega) : \Delta u \in L^2(\Omega)\},$$ with the norm $$\|u\|_{H^1_\Delta(\Omega)}^2 := \|u\|_{H^1(\Omega)}^2+\|\Delta u\|_{L^2(\Omega)}^2.$$

I'm wondering if the norms $\|u\|_{H_\Delta(\Omega)}$ and $\|u\|_{H^1_\Delta(\Omega)}$ are equivalent to $\|u\|_{H^2(\Omega)}$. What about the trace theorem in the spaces $H_\Delta(\Omega)$ and $H^1_\Delta(\Omega)$ ? Thank you.

Answer 1

They not the same unless if you work with Dirichlet boundary conditions.