Given a domain $\Omega \subset R^n$, consider the set $D := \{u \in L^2(\Omega)| \Delta u \in L^2(\Omega)\}$, where $-\Delta$ is the Laplacian. I think this is the domain of the adjoint of $-\Delta$.

My question: is the set $D$ always identical to the Sobolev space $H^2(\Omega)$ ?