= Clamped Hessian identity
{title2=$\|D^2u\|_2^2=\|\Delta u\|_2^2$}
For $u\in H_0^2(U)$ on an <open set> $U$, the identity $\sum_{i,j}\|\partial_{ij}u\|_2^2=\|\Delta u\|_2^2$ follows by two <integrations by parts> for compactly supported <test functions>, followed by $H^2$ density. Thus the <Laplacian> and full <Hessian> have the same <L2 norm> on the <clamped second-order Sobolev space>.
If $U$ is bounded, the <Poincaré inequality> applies to $u$ and its first <weak derivatives>, all of which have zero-boundary approximations. It then controls the full $H^2$ <Sobolev norm> by $\|\Delta u\|_2$. The exact second-derivative identity holds without this boundedness hypothesis; the full inhomogeneous norm estimate needs the extra low-frequency control.
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