Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 107 / 2 / v / 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 Ω \in C^{2, α}, ϕ \in C^{2, α} (\overline{Ω}), f \in C^{0, α} (\overline{Ω}),

with

a^{ij}, b^{i}, c \in C^{0, α} (\overline{Ω})

and uniform ellipticity. Then the Global Schauder estimate gives

u \in C^{2, α} (\overline{Ω})

and bounds its norm by the forcing, boundary data, and

C^{0}

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