= Solution
First perform <homogenization of Dirichlet boundary data>: put $v=u-\psi\in H_0^1(B_1^+)$ and $g=f-\Delta\psi\in L^2(B_1^+)$. Then $\Delta v=g$ weakly. Extend both functions oddly across the flat face:
$$
\widetilde v(x',t)=\begin{cases}v(x',t)&t>0,\\-v(x',-t)&t<0,\end{cases}\qquad\widetilde g(x',t)=\begin{cases}g(x',t)&t>0,\\-g(x',-t)&t<0.\end{cases}
$$
The zero <Sobolev trace> of $v$ makes this <odd reflection> an $H^1(B_1)$ function. This can also be seen by approximating $v$ by smooth <compactly supported> functions in $B_1^+$ and reflecting the approximants. Its $H^1$ <norm> is $\sqrt2$ times that of $v$, and $\|\widetilde g\|_{L^2(B_1)}=\sqrt2\|g\|_{L^2(B_1^+)}$.
For $\phi\in C_c^\infty(B_1)$, the function $\phi(x',t)-\phi(x',-t)$ on the upper half-ball has zero trace on the flat face and vanishes near the curved face, hence is an admissible $H_0^1$ test. Splitting the integrals over the two half-balls and changing $t$ to $-t$ shows
$$
\int_{B_1}D\widetilde v\cdot D\phi=-\int_{B_1}\widetilde g\phi.
$$
Thus $\Delta\widetilde v=\widetilde g$ weakly throughout the full ball; there is no extra distribution on the reflecting face.
Apply part (i) to $B_{1/2}\Subset B_1$, restrict to the upper half-ball, and then add $\psi$. Since $\|\Delta\psi\|_2\leq C(n)\|\psi\|_{H^2}$, this gives $u\in W^{2,2}(B_{1/2}^+)$ and the <second-derivative estimate at a flat Dirichlet boundary>
$$
\boxed{\|u\|_{W^{2,2}(B_{1/2}^+)}\leq C(n)\left(\|u\|_{W^{1,2}(B_1^+)}+\|f\|_{L^2(B_1^+)}+\|\psi\|_{W^{2,2}(B_1^+)}\right).}
$$
The fixed radii leave distance $1/2$ for the interior estimate, so the constant depends only on dimension.
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