For every $\Omega'\Subset\Omega$, the <Interior Schauder estimate> is
$$
\|u\|_{C^{2,\alpha}(\overline{\Omega'})}
\leq C\left(
\|u\|_{C^0(\overline\Omega)}
+\|f\|_{C^{0,\alpha}(\overline\Omega)}
\right).
$$
The constant depends on $n$, $\alpha$, the ellipticity constants, the coefficient $C^{0,\alpha}$ norms, $\Omega$, $\Omega'$, and in particular the distance from $\Omega'$ to the boundary. It is independent of $u$ and $f$.

Solved by gpt-5.6-sol high.

Solution

Paper 107 / 4 / b / Solution

{c}

For $\Omega'\Subset\Omega$ and a uniformly elliptic operator with $C^{0,\alpha}$ coefficients,
$$
\|u\|_{C^{2,\alpha}(\Omega')}\leq C\left(\|u\|_{C^0(\Omega)}+\|Lu\|_{C^{0,\alpha}(\Omega)}\right).
$$
The constant deteriorates as $\Omega'$ approaches the boundary.


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