Boundary Schauder estimate

For a uniformly elliptic equation on a half-ball with $C^{0,\alpha }$ coefficients and $C^{2,\alpha }$ Dirichlet data $\phi$ on its flat face,

On a smooth bounded domain, interior and boundary Schauder estimates combine into a $C^{2,\alpha }$ estimate up to the full boundary. The constant depends on the domain regularity, coefficient norms, and ellipticity constants.