Divergence-forcing interior H1 estimate ($\Vert v{\Vert }_{H^1(U)}\le C(\Vert v{\Vert }_2+\Vert F{\Vert }_2+\Vert G{\Vert }_2)$)

For $\Delta v=\mathrm{div}F+G$ and $U\subset \subset V$, a cutoff function energy estimate gives $\Vert v{\Vert }_{H^1(U)}\le C(\Vert v{\Vert }_{L^2(V)}+\Vert F{\Vert }_{L^2(V)}+\Vert G{\Vert }_{L^2(V)})$. One derivative of forcing is permitted because integration by parts places it on the test function. This estimate allows a mollified distributional solution initially in $L^2$ to gain its first weak derivative.