Consider the Laplacian $\Delta$ as an operator on $L^2(\mathbb R^n)$, densely defined on the subspace $C^\infty_0(\mathbb R^n)$.
Is the domain of the closure of the Laplacian, in the sense described here: https://en.wikipedia.org/wiki/Unbounded_operator#Closed_linear_operators, equal exactly to: $\{u \in L^2(\mathbb R^n) | \Delta u \in L^2(\mathbb R^n)\}$ (where $\Delta$ here means in the distributional sense)?
Does any of the above spaces (which I hope are equal) in turn exactly equal the Sobolev space $W^{2,2}(\mathbb R^n)$, or is $W^{2,2}$ actually a strictly smaller space?
Does any of the above spaces equal the Friedrichs extension?