= Clamped Hessian identity
{title2=$\|D^2u\|_2^2=\|\Delta u\|_2^2$}
For $u\in H_0^2(U)$, $\sum_{i,j}\|\partial_{ij}u\|_2^2=\|\Delta u\|_2^2$. Two <integrations by parts> prove this for compactly supported <test functions>, and $H^2$ density extends it to the <clamped second-order Sobolev space>. Together with the <Poincare-Wirtinger inequality> applied to the mean-zero first derivatives and the zero-boundary <Poincare inequality>, it controls the full $H^2$ <norm> by $\|\Delta u\|_2$. The boundary conditions are essential to this exact identity on bounded domains.
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