= Second-derivative estimate at a flat Dirichlet boundary
{title2=$\|u\|_{W^{2,2}(B_1^+)}\leq C(\|u\|_{W^{1,2}(B_2^+)}+\|f\|_2)$}
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.
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