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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 107 / 2 / v / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-107/2/v/solution