Let us define the Laplace operator $A:D(A)\subset L^2(\Omega)\to L^2(\Omega)$ by setting $Au:=\Delta u$ and $D(A):=H^2(\Omega)$ (without any boundary condition), where $\Omega$ is a bounded domain with smooth boundary. From Weyl's lemma, we can infer that this operator is closable. However, the proposition 4.1 in the paper, titiled "Boundary value problems for elliptic partial differential operators on bounded domains", shows that $A$ is not closed. The approach there is not direct and has to use the concept of "quasiboundary triple". I just wonder if there is a simple, direct method to show that $A$ is not closed.

$\begingroup$ In the onedimensional case of $\Delta$ on a finite interval $[a,b]$, it is easy to see that there is a continuum of (proper) selfadjoint extensions beyond $H^2$, depending on choices of boundary conditions. $\endgroup$– paul garrettNov 26, 2016 at 16:34
1 Answer
In the multidimensional case the domain of the maximal operator associated to $\Delta$ in $L^2(\Omega)$ is given by $$ D_{\max}:=\{f\in L^2(\Omega):\Delta f \in L^2(\Omega)\}, $$ and here $\Delta f$ is understood in the sense of distributions. This space is larger than $H^2(\Omega)$ as e.g. all harmonic functions in $L^2(\Omega)$ belong to $D_{max}$, but these harmonic functions are not necessarily in $H^2(\Omega)$ (since their weak derivatives may not be square integrable).
Now equip $D_{max}$ with the graph norm $\Vert f\Vert:=\Vert f\Vert_{L^2} +\Vert\Delta f\Vert_{L^2}$. It is known that the space $C^\infty(\overline\Omega)$ is dense in $D_{max}$ and hence also $H^2(\Omega)$ is dense in $D_{max}$ with respect to the graph norm, but $H^2(\Omega)\not=D_{max}$ by the argument above. The maximal operator associated to $\Delta$ coincides with the adjoint of the minimal operator defined on $H^2_0(\Omega)$ and hence the maximal operator is closed. It follows that the closure of the Laplacian defined on $H^2(\Omega)$ is the maximal operator and that the Laplacian defined on $H^2(\Omega)$ is not closed.
In the onedimensional case the situation is different: If $\Omega=(a,b)$, then the maximal domain $D_{max}$ coincides with $H^2((a,b))$, and hence in this case the Laplacian $\Delta=\frac{d^2}{dx^2}$ defined on $H^2((a,b))$ is closed.

$\begingroup$ Thank you very much. The argument is fine and acceptable. But could you give an example of a distribution that $u\in D_{max}$ but not in $H^2$? I need to explain to my students. So, an easy one is appreciated. $\endgroup$– Ice seaNov 27, 2016 at 5:22

$\begingroup$ I think a harmonic function with the form $\frac{1}{x1^\alpha}$ may work. Thank you very much $\endgroup$– Ice seaNov 27, 2016 at 6:50

$\begingroup$ Does this answer contradict the comment by @paulgarret under the question? $\endgroup$– BananachMay 5, 2018 at 7:34