Uniform boundary Schauder bootstrap after a C1-Holder seed

Freeze the realized coefficients along each solution. The seed makes $A(x,u,Du)$ and $b(x,u,Du)$ uniformly $C^{0,\beta }$, so [F2] gives a common $C^{2,\beta }$ bound on a smaller patch. Each candidate is individually smooth, satisfying the classical regularity prerequisite of that estimate. Interior balls use its interior counterpart.

Suppose a uniform $C^{m,\beta }$ bound is known, $m\ge 2$. Differentiate the equation $m-1$ times in flat tangential directions. For $z=D_T^{m-1}u$, the principal term is $A^{ij}{z}_{ij}$. The chain-rule terms containing $Dz$ enter bounded $C^{0,\beta }$ first-order coefficients; every remaining term involves derivatives of $u$ of order at most $m$, hence has a uniform $C^{0,\beta }$ bound. Since the zero trace differentiates tangentially, $z=0$ on the flat boundary. [F2] bounds $z$ in $C^{2,\beta }$ on a smaller patch. This controls all derivatives of order $m+1$ with at most two normal directions.

Solve the original equation for $u_{nn}$, using $A^{nn}\ge \lambda >0$, and differentiate it $m-1$ times. Recover the derivatives of order $m+1$ by increasing their number of normal indices: every other second-derivative slot has at least one tangential index and therefore fewer normal indices than the derivative currently solved for; derivatives of the coefficients involve only derivatives of $u$ through order $m$. Thus the already controlled derivatives and the previous normal-count stages give all remaining $C^{0,\beta }$ bounds. Iteration in $m$ on nested patches with a fixed smaller limiting patch proves all finite orders. A further bounded derivative implies the desired $C^{k,\alpha }$ bound for any $\alpha <1$: in a fixed convex flattened half-ball or interior ball, apply [F3] to $t\mapsto D^{\gamma }u(x+t(y-x))$, $0\le t\le 1$, $\mathrm{\mid }\gamma \mathrm{\mid }=k$. The chain rule and scalar mean value theorem give $\mathrm{\mid }{D}^{\gamma }u(y)-D^{\gamma }u(x)\mathrm{\mid }\le C\mathrm{\parallel }{D}^{k+1}u{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }y-x\mathrm{\mid }$, including boundary endpoints by continuity. Dividing by $\mathrm{\mid }y-x{\mathrm{\mid }}^{\alpha }$ bounds the seminorm by $C\mathrm{\parallel }{D}^{k+1}u{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{diam}(B{)}^{1-\alpha }$; fixed smooth chart changes preserve these bounds by the same chain rule. A finite cover gives the estimate on the prescribed smaller patch: nearby pairs lie in a common chart, and separated pairs use the already bounded derivative suprema.