Clamped Hessian identity ($\Vert D^{2}u{\Vert }_2^2=\Vert \Delta u{\Vert }_2^2$)

For $u\in H_0^2(U)$ on an open set $U$, the identity ${\sum }_{i,j}\Vert {\partial }_{ij}u{\Vert }_2^2=\Vert \Delta u{\Vert }_2^2$ 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 $\Vert \Delta u{\Vert }_2$. The exact second-derivative identity holds without this boundedness hypothesis; the full inhomogeneous norm estimate needs the extra low-frequency control.