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Let $d=\mathrm{dist}({\Omega }^{\prime },\partial \Omega )>0$. Choose a smooth cutoff function $\eta \in C_c^{\infty }(\Omega )$ with $0\le \eta \le 1$, $\eta =1$ on ${\Omega }^{\prime }$ and $|D\eta |\le C(n)/d$. Such cutoffs can be constructed by mollifying the indicator of a fixed small neighbourhood of $\overline{{\Omega }^{\prime }}$. Write $M=\Vert A{\Vert }_{\infty ,\mathrm{op}}$, $B=\Vert b{\Vert }_{\infty }$ and $C_0=\Vert c{\Vert }_{\infty }$; these are controlled by the individual coefficient bounds and dimension.

The weak formulation, extended to compactly supported $H^1$ tests by density, isWe may use $\phi ={\eta }^{2}u$, since $u\in H_{\mathrm{loc}}^1$. Expanding its gradient and using uniform ellipticity givesApply Young inequality to the first two terms, allocating at most $\lambda /4$ of the gradient integral to each. Bound $|fu|\le (f^2+u^2)/2$ in the last term. Absorbing those gradient contributions yields the Caccioppoli inequality with bounded lower-order termsConsequently $\Vert Du{\Vert }_{L^2({\Omega }^{\prime })}\le C(\Vert u{\Vert }_{L^2(\Omega )}+\Vert f{\Vert }_{L^2(\Omega )})$. Adding the $L^2$ norm of $u$ proveswhere $C$ has precisely the stated dependence on dimension, ellipticity, coefficient bounds and $d$. The proof uses $|c|$, so no sign condition on the zeroth-order term is required.