Operator theory of the Hessian How can I learn more about the operator theory of the Hessian? The Hessian of a function $u : \Omega \rightarrow \mathbb R$ over a domain $\Omega \subseteq \mathbb R^n$ is the matrix of second derivatives $D^2 u$.
The Hessian can be seen as an unbounded linear operator $D^2 : X \subseteq L^2(\Omega,\mathbb R) \rightarrow L^2(\Omega, \mathbb R^{n \times n})$,
where $X$ denotes the domain of the unbounded operator.
The natural question is then which choices of the domain yield a densely-defined closed Hessian. Note that the maximal choice of domain is the Sobolev space $H^2$, and I am particularly interested about "reasonable" boundary conditions.
 A: If  you have a closed, densely defined  operator $A: D(A)\subset H_0\to H_1$, $H_0, H_1$, Hilbert spaces,  then you can define on $D(A)$ the graph norm
$$\Vert u\Vert_A:=\Vert u\Vert_{H_0} +\Vert Au\Vert_{H_1}. $$
The space $D(A)$ with the above norm is a Banach space.
If $X\subset D(A)$, then the restriction of $A$ to $X$ is closed  iff  $X$ is closed in $D(A)$ with respect to the norm $\Vert-\Vert_A$.
In the case at hand we take $\newcommand{\bR}{\mathbb{R}}$
$$H_0= L^2(\Omega,\bR),\;\; H_1:= L^2(\Omega,\bR^{n\times n}), $$
$$D(A)= H^2(\Omega),\;\; Au=D^2 u. $$
In this case the norm $\Vert-\Vert_A$ is equivalent to the norm  $\Vert-\Vert_{H^2(\Omega)}$ (think interpolation inequalities.)
Finding "boundary conditions"   so that the corresponding operator is closed and densely defined is equivalent to finding  closed subspaces of $H^2(\Omega)$ that are dense in $L^2(\Omega)$.  Observe that any closed subspace of $H^2(\Omega)$  can be described asa the kernel of a bounded operator $B: H^2(\Omega)\to H$, where  $H$ is some Hilbert space. (The operator $B$ gives the "boundary" condition $Bu=0$. )
