Ellipticity recovers every remaining normal derivative

Work near an arbitrary point of the flat boundary. By continuity of $J(x)=(x,w(x),Dw(x))$ and the smooth local extensions, choose a smaller closed convex coordinate half-box $K$ whose jet image lies in a closed convex box inside the coefficient extension domain. Every coefficient derivative through order $m$ is bounded on this jet box by F8. On $K$, derivatives of $w$ through order $m$ are bounded, and the order-$m$ fields are $\alpha$-Hölder by the hypothesis. For each derivative of order at most $m-1$, apply F6 along successive interior coordinate segments and then pass to their boundary limits: its increment is bounded by a constant times $\mathrm{\mid }x-y\mathrm{\mid }$. On a bounded box, $\mathrm{\mid }x-y\mathrm{\mid }\le (\mathrm{diam}K{)}^{1-\alpha }\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }$. Thus all derivatives of $J$ through order $m-1$ are bounded and $\alpha$-Hölder on $K$. Shrink $K$ once more if needed so that F2 holds on it.

For bounded $\alpha$-Hölder functions on $K$, direct subtraction gives $[fg{]}_{0,\alpha }\le \mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}[g{]}_{0,\alpha }+\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}[f{]}_{0,\alpha }$. If $f\ge \lambda >0$, then $\mathrm{\parallel }1\mathrm{/}f{\mathrm{\parallel }}_{\mathrm{\infty }}\le {\lambda }^{-1}$ and $[1\mathrm{/}f{]}_{0,\alpha }\le {\lambda }^{-2}[f{]}_{0,\alpha }$, since $\mathrm{\mid }1\mathrm{/}f(x)-1\mathrm{/}f(y)\mathrm{\mid }=\mathrm{\mid }f(x)-f(y)\mathrm{\mid }\mathrm{/}\mathrm{\mid }f(x)f(y)\mathrm{\mid }$. Finite sums and finite products therefore preserve $C^{0,\alpha }$. Also, any smooth coefficient derivative through order $m-1$ is Lipschitz on the convex jet box by F6 and the order-$m$ bounds from step 1.1; composing it with the Lipschitz map $J$ gives a bounded $\alpha$-Hölder function on $K$.

Set $a^{ij}(x)=A^{ij}(J(x))$ and $b(x)=B(J(x))$. Repeated chain and coordinate product rules express every interior derivative of these functions through order $m-1$ as a finite sum of terms: a coefficient derivative of order at most $m-1$ evaluated at $J$, multiplied by derivatives of components of $J$ whose total differentiation order is at most $m-1$. Every factor has a bounded $\alpha$-Hölder extension by steps 1.1 and 2.1, so every such term does too. Hence $a^{ij}$ and $b$ belong to $C^{m-1,\alpha }(K)$. In particular $a^{nn}\ge \lambda$ and its reciprocal is bounded and $\alpha$-Hölder by step 2.1.

Fix a multi-index $\beta$ with $\mathrm{\mid }\beta \mathrm{\mid }=m+1$ and ${\beta }_n=k\ge 3$, and put $\gamma =\beta -2e_n$, where $e_n$ is the last coordinate unit multi-index. In the smooth interior, differentiate the assumed equation $\mathrm{\mid }\gamma \mathrm{\mid }=m-1$ times. By F5 and F7, isolating the undifferentiated normal-normal coefficient gives $a^{nn}{D}^{\beta }w=-D^{\gamma }b-{\sum }_{(i,j)\mathrm{\ne }(n,n)}{a}^{ij}{D}^{\gamma }{w}_{ij}-{\sum }_{i,j}{\sum }_{0<\eta \le \gamma }{c}_{\gamma \eta }(D^{\eta }{a}^{ij})(D^{\gamma -\eta }{w}_{ij})$, with $c_{\gamma \eta }={\prod }_{\mathrm{\ell }=1}^n{\gamma }_{\mathrm{\ell }}!\mathrm{/}({\eta }_{\mathrm{\ell }}!({\gamma }_{\mathrm{\ell }}-{\eta }_{\mathrm{\ell }})!)$ and componentwise $\eta \le \gamma$. In the last sum, the derivative of $w$ has order $m+1-\mathrm{\mid }\eta \mathrm{\mid }\le m$; $D^{\gamma }b$ involves derivatives of $w$ of order at most $m$. These terms have bounded $\alpha$-Hölder extensions by steps 1.1, 2.1 and 3.1 and the given regularity. Each derivative in the middle sum has total order $m+1$ and at most ${\gamma }_n+1=k-1$ normal differentiations.

Start with $k\le 2$, supplied by F2, and induct on $k$ up to $m+1$. Assuming every total-order $m+1$ derivative with fewer than $k$ normal differentiations has a bounded $\alpha$-Hölder extension to $K$, the middle sum in step 4.1 is controlled by that assumption. All remaining right-hand terms are already controlled independently of the induction. Divide by $a^{nn}\ge \lambda$ using step 2.1 to obtain a bounded $\alpha$-Hölder extension of $D^{\beta }w$. The formula agrees with the classical derivative in the interior and so defines its boundary values uniquely. There are finitely many derivatives at this fixed order, and the same $K$ suffices for all of them. This proves the claimed control of every order-$m+1$ derivative.

Compatibility with the already continuous order-$m$ derivatives follows directly from F6. Write $v=D^{\mu }w$ with $\mathrm{\mid }\mu \mathrm{\mid }=m$ and let $g_j$ be the continuous extension of ${\mathrm{\partial }}_{j}v$ obtained in step 5.1. At a flat-boundary point $x$, apply F6 to the tangential segment from $x+\epsilon e_n$ to $x+h{e}_j+\epsilon e_n$, for $j<n$ and $\epsilon >0$. Subtract $g_j(x)$ from the resulting difference quotient and bound its absolute value by the supremum of $\mathrm{\mid }{g}_j(y)-g_j(x)\mathrm{\mid }$ over that segment. Let $\epsilon$ decrease to zero using continuity of $v$, and then let $h$ tend to zero using continuity of $g_j$. The boundary tangential derivative of $v$ is therefore $g_j(x)$. For the inward normal direction, F6 applies directly to $v$ on the closed segment from $x$ to $x+h{e}_n$, since $v$ is continuous at its endpoint and smooth in its interior; the one-sided quotient tends to $g_n(x)$. The order-$m+1$ extensions from step 5.1 and the lower-order extensions in the hypothesis consequently satisfy the boundary-extension convention of F3. Thus $w\in C^{m+1,\alpha }$ on the smaller patch, without prescribing any normal boundary values.