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Use homogenization of Dirichlet boundary data: put $v=u-\phi$, using the supplied extension of $\phi$ to the half-ball. Then $v=0$ on the flat face and $\Delta v=f-\Delta \phi$. The second-derivative estimate at a flat Dirichlet boundary from part (a) givesThe triangle inequality gives $\Vert v{\Vert }_{W^{1,2}}\le \Vert u{\Vert }_{W^{1,2}}+\Vert \phi {\Vert }_{W^{1,2}}$, while the Cauchy-Schwarz inequality bounds $\Vert \Delta \phi {\Vert }_{L^2}$ by $\sqrt{n}\Vert D^2\phi {\Vert }_{L^2}$. Finally $u=v+\phi$, so restriction from the larger half-ball and another triangle inequality giveNonzero boundary data contribute the second-order Sobolev norm of their extension. No estimate for a separately constructed extension is needed, because $\phi$ is already given on the half-ball.

Use the Poisson integral to choose a smooth harmonic function $G$ on the unit disk whose boundary value is $g$. Seek $v$ in the form $v=w+G$, which is the standard homogenization of Dirichlet boundary data. Then $w\in H_0^1(\Omega )$ and, since $\Delta G=0$,Part (d) gives a unique such $w$, and henceis the unique weak solution with boundary value $g$.

For a classical solution of $\Delta u=f$ on a half-ball with zero data on its flat face, a tangential Caccioppoli inequality controls every second partial derivative having at least one tangential index. The equation recovers the remaining second normal derivative. The resulting Sobolev norm estimate has a dimension-only constant for the fixed radii one and two. Nonzero data are handled by homogenization of Dirichlet boundary data, adding the $W^{2,2}$ norm of the supplied boundary extension. No condition is imposed on the curved face because the smooth cutoff function vanishes there.