Given a domain $\Omega \subset\Bbb R^n$, 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 $H^2(\Omega)$ ?

Perhaps this is dependent on the boundedness or smoothness of $\Omega$. I didn't found anything about this problem in functional analysis textbooks. Are such sources known to anybody ?