= Solution
The stated assumptions alone do not imply regularity up to the boundary: continuous boundary data need not have two Hölder derivatives. A standard sufficient set of hypotheses for the <Global Schauder estimate> is
$$
\partial\Omega\in C^{2,\alpha},\qquad
\phi\in C^{2,\alpha}(\overline\Omega),\qquad
f\in C^{0,\alpha}(\overline\Omega),
$$
with $a^{ij},b^i,c\in C^{0,\alpha}(\overline\Omega)$ and uniform ellipticity. Then the <Global Schauder estimate> gives
$$
\boxed{u\in C^{2,\alpha}(\overline\Omega)}
$$
and bounds its norm by the forcing, boundary data, and $C^0$ norm.
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