 Solution

Failure of the bound would give normalized $v_j$ with $\Vert v_j{\Vert }_{\infty }=1$, zero Dirichlet boundary condition, and $\Vert L{v}_j{\Vert }_{C^{0,\alpha }}\to 0$. The global Schauder estimate bounds them in $C^{2,\alpha }$. By compact embedding of Hölder spaces, a subsequence converges in $C^{2,\beta }$ to $v$ with $Lv=0$, zero boundary data and $\Vert v{\Vert }_{\infty }=1$. The uniform Hölder seminorm bounds pass to the limit, so $v\in C^{2,\alpha }$. This contradicts the trivial null space, provingThe global Schauder estimate then gives $\Vert u{\Vert }_{C^{2,\alpha }}\le K\Vert f{\Vert }_{C^{0,\alpha }}$, the bounded-inverse estimate used next.