First, let us define the following space : \begin{align*} H^{1}(\Omega) :=\{u :\Omega\to\mathbb{R}\,|\,\forall\alpha\text{ multiindex },0\leq|\alpha|\leq1,||D^{\alpha}u||_{L^{2}(\Omega)}<\infty\} \\ H^{2}(\Omega) :=\{u :\Omega\to\mathbb{R}\,|\,\forall\alpha\text{ multiindex },0\leq|\alpha|\leq2,||D^{\alpha}u||_{L^{2}(\Omega)}<\infty\}\\ \end{align*} Then, we can define the corresponding space $H_{0}^{1}(\Omega) \subset H^{1}(\Omega)$ and $H_{0}^{2}(\Omega)\subset H^{2}(\Omega)$.
Finally we define the following : \begin{equation*} \begin{cases} D(B) := \{u \in H_{0}^{1}(\Omega)\,|\,\Delta u\in L^{2}(\Omega)\} \\ \forall u\in D(B), B u=\Delta u \end{cases} \end{equation*}
I can immediately see by definition that $D(B)\subset H^{2}(\Omega)$. However, can anyone help me to construct a counter example or to show that $H_{0}^{2}(\Omega) \subset D(B)$?
Any help is much appreciated! Thank you very much!