= Interior second-derivative estimate for the Poisson equation
{c}
{title2=$\|u\|_{H^2(U)}\leq C(\|u\|_{H^1(V)}+\|\Delta u\|_{L^2(V)})$}
If $u\in H^1(V)$ and $\Delta u=f\in L^2(V)$ weakly, then $u\in H^2_{\mathrm{loc}}(V)$. Test the weak equation with $-\delta_{-h}(\eta^2\delta_hu)$, move the first difference quotient onto $Du$, and bound the remaining quotient on the forcing side by the first derivative of $\eta^2\delta_hu$. <Young inequality> absorbs the localized second derivatives. Uniform bounds in $h$, together with the <Sobolev characterization by bounded difference quotients>, give the assertion without differentiating $f$. The constant depends only on dimension and the positive distance from $U$ to the boundary of $V$.
// Target: geometry-and-topology.bigb
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