= Solution
Choose a <smooth cutoff function> $\eta$ supported in $B_2$, equal to one on $B_1$, with $0\leq\eta\leq1$ and $|D\eta|\leq C$. For $i<n$, put $v_i=D_i u$. The zero <Dirichlet boundary condition> implies that the <tangential boundary derivative> $v_i$ vanishes on the flat face. Differentiating the <Poisson equation> gives $\Delta v_i=D_i f$. Test this equation with $\eta^2v_i$. The boundary terms vanish because $v_i=0$ on the flat face and $\eta=0$ near the curved face. <Integration by parts> in the tangential direction moves $D_i$ off $f$, yielding
$$
\int_{B_2^+}\eta^2|Dv_i|^2
=\int_{B_2^+}fD_i(\eta^2v_i)
-2\int_{B_2^+}\eta v_i Dv_i\cdot D\eta.
$$
This is the essential <Caccioppoli inequality> step: it involves $f$, rather than $Df$. Expanding $D_i(\eta^2v_i)$ and using the <Cauchy-Schwarz inequality> and the <Young inequality> gives
$$
\int\eta^2|Dv_i|^2
\leq\frac12\int\eta^2|Dv_i|^2
+C\int\bigl(f^2+v_i^2|D\eta|^2\bigr).
$$
The term $f\eta(D_i\eta)v_i$ is bounded by $C f^2+C|D\eta|^2v_i^2$. Absorbing the first term and summing over $i<n$ proves the hinted estimate:
$$
\sum_{i<n}\sum_{j=1}^n\|D_{ij}u\|_{L^2(B_1^+)}^2
\leq C(n)\left(\|Du\|_{L^2(B_2^+)}^2+\|f\|_{L^2(B_2^+)}^2\right).
$$
All mixed second <partial derivatives> are now controlled, since $D_{ni}u=D_{in}u$. The remaining <normal derivative> is recovered from the equation:
$$
D_{nn}u=f-\sum_{i<n}D_{ii}u.
$$
The <Cauchy-Schwarz inequality> bounds its squared <L2 norm> by $n$ times the sum of the squared <L2 norms> on the right. Including the already controlled zeroth and first <partial derivatives>, and taking square roots, gives the <second-derivative estimate at a flat Dirichlet boundary>:
$$
\boxed{\|u\|_{W^{2,2}(B_1^+)}\leq C(n)\left(\|u\|_{W^{1,2}(B_2^+)}+\|f\|_{L^2(B_2^+)}\right).}
$$
\b[The full second-order Sobolev norm is controlled without any norm of $Df$.]
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