= Divergence-forcing interior H1 estimate
{title2=$\|v\|_{H^1(U)}\leq C(\|v\|_2+\|F\|_2+\|G\|_2)$}
For $\Delta v=\operatorname{div}F+G$ and $U\subset\subset V$, a <cutoff function> energy estimate gives $\|v\|_{H^1(U)}\leq C(\|v\|_{L^2(V)}+\|F\|_{L^2(V)}+\|G\|_{L^2(V)})$. 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>.
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