 Interior second-derivative estimate for the Poisson equation

If $u\in H^1(V)$ and $\Delta u=f\in L^2(V)$ weakly, then $u\in H_{\mathrm{loc}}^2(V)$. Test the weak equation with $-{\delta }_{-h}({\eta }^2{\delta }_{h}u)$, move the first difference quotient onto $Du$, and bound the remaining quotient on the forcing side by the first derivative of ${\eta }^2{\delta }_{h}u$. 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