Solution
= Solution
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.