Let $\Omega \subset \mathbb R^d$ be a bounded open set. Let $H_0^1(\Omega)$ be the usual Sobolev space.
From the definition it follows, that each $u \in H_0^1(\Omega)$ can be approximated by a sequence $\{u_n\} \subset C_c^\infty(\Omega)$ (smooth and compactly supported functions).
Is it possible to somehow control the higher derivatives of the approximating sequence? Of course, we must have $\|u_n\|_{H^2(\Omega)} \to \infty$ if $u \not\in H^2(\Omega)$. But is it, e.g., possible to obtain a sequence $\{u_n\} \subset H^2(\Omega) \cap H_0^1(\Omega)$ with $$\|u - u_n\|_{L^2(\Omega)} \le \frac{C}n \quad\text{and}\quad \|u_n\|_{H^2(\Omega)} \le C \, n\quad?$$
One possibility would be to extend $u$ by zero and use a mollification argument. This should yield a sequence $\{u_n\}\subset H^2(\Omega)$ satisfying the above properties. However, I do not see a reason how to construct the approximating sequence in $H_0^1(\Omega)$.