= Solution
Put $\delta=\operatorname{dist}(\Omega',\partial\Omega)>0$. Choose a smooth cutoff $0\leq\eta\leq1$, equal to one on $\Omega'$, supported within its $\delta/2$ neighborhood, with $|D\eta|\leq C_n/\delta$. Such a cutoff is obtained by smoothing a distance cutoff; its support is <relatively compact> in $\Omega$. Let $0<|h|<\delta/8$, fix a coordinate $k$, and put $z=\Delta_k^h u$. It lies in $W^{1,2}$ on the cutoff neighborhood because both translated copies of $u$ do. Extend $\eta^2z$ by zero, and use the <test function>
$$
\phi=-\Delta_k^{-h}(\eta^2z)\in W_0^{1,2}(\Omega).
$$
The weak equation is $\int Du\cdot D\phi=-\int f\phi$. The discrete integration-by-parts identity $\int a\Delta_k^{-h}b=-\int(\Delta_k^h a)b$, valid by translation with these supports, turns it into
$$
\int Dz\cdot D(\eta^2z)=\int f\Delta_k^{-h}(\eta^2z).
$$
Write $X=\|\eta Dz\|_2$, $U=\|z\|_{L^2(\operatorname{supp}\eta)}$, $A=\|D\eta\|_\infty$ and $F=\|f\|_2$. Expanding the left side gives $X^2$ plus a cross term of absolute value at most $2AUX$. Applying part (a) on the whole space to the zero extension of $\eta^2z$ bounds the right side by
$$
F\|D_k(\eta^2z)\|_2\leq F(X+2AU).
$$
Consequently
$$
X^2\leq(2AU+F)X+2AUF\leq\tfrac12X^2+C(A^2U^2+F^2),
$$
where the last step is the elementary <Young inequality>. Part (a), on a slightly larger cutoff neighborhood if needed, gives $U\leq\|D_k u\|_{L^2(\Omega)}$. We conclude that
$$
\|D\Delta_k^h u\|_{L^2(\Omega')}\leq C_n\left(\|f\|_{L^2(\Omega)}+\delta^{-1}\|Du\|_{L^2(\Omega)}\right)
$$
uniformly as $h\to0$.
To obtain actual second <weak derivatives>, choose $h_j\to0$. For each $i,k$, the bounded sequence $\Delta_k^{h_j}D_i u$ has a <weakly convergent> subsequence in $L^2(\Omega')$, say to $g_{ik}$. For every smooth compactly supported $\psi$ in $\Omega'$,
$$
\int g_{ik}\psi=\lim_j\int(\Delta_k^{h_j}D_i u)\psi
=-\lim_j\int D_i u\,\Delta_k^{-h_j}\psi=-\int D_i uD_k\psi.
$$
Thus $g_{ik}=D_kD_i u$ is the second <weak derivative>. <weak lower semicontinuity> gives the same uniform bound for these derivatives. Adding the already known zeroth and first derivative norms and summing over the finitely many coordinates proves
$$
\boxed{u\in W_{\mathrm{loc}}^{2,2}(\Omega),\qquad
\|u\|_{W^{2,2}(\Omega')}\leq C(n,\delta)\bigl(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^2(\Omega)}\bigr).}
$$
All derivative existence and estimates have been established by the difference-quotient argument; no second derivatives were assumed in choosing the test. This proves the <Interior second-derivative estimate for the Poisson equation> in the stated setting.
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