= Solution
Take $\Omega'$ nonempty. If the bound failed, normalization would give nonnegative $v_j$ with supremum one, zero boundary values, $\inf_{\Omega'}v_j\to0$, and $\|Lv_j\|_{C^{0,\alpha}}\to0$. The <global Schauder estimate> bounds them in the <Hölder space> $C^{2,\alpha}$. By <compact embedding of Hölder spaces>, a subsequence converges in $C^{2,\beta}$, $0<\beta<\alpha$, to $v\geq0$ with $Lv=0$, zero boundary data and supremum one.
Points approaching the infima have a subsequence converging in $\overline{\Omega'}\subset\Omega$. Uniform convergence gives an interior zero of $v$, and part (c), the <strong minimum principle for elliptic operators> after replacing $c$ by $\min(c,0)$, gives $v=0$. This contradicts its supremum one, proving
$$
\boxed{\sup_\Omega u\leq C\bigl(\inf_{\Omega'}u+\|f\|_{C^{0,\alpha}}\bigr).}
$$
Back to article page
