Let $U\subset\mathbb{R}^n$ be open, and let $V\subsetneq W\subsetneq U$ be open sets with $\overline V$ compact and contained in $W$ and $\overline W$ compact and contained in $U$. Let $u\in H^1(U)$ satisfy
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\begin{align*}
-\Delta u=f
\end{align*}
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in the weak sense on $U$, where $f\in L^2(U)$. Then $u\in H^2(V)$ and there exists $C>0$, depending on $n$ and on the cutoff geometry needed to localize from $W$ to $V$, in particular on separation distances such as $\operatorname{dist}(V,\partial W)$, such that