 Solution

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.