Classical C4-alpha maximal graphs with fixed smooth boundary are smooth

Fix an interior point and suppose $w$ is $C^{m,\alpha }$ on a ball about it, where $m\ge 4$. For every multi-index $I$ of length $m-2$ and each coordinate direction $e_j$, put $v=D^{I}w$ and $q_h=(v(x+h{e}_j)-v(x))\mathrm{/}h$. For each fixed nonzero sufficiently small $h$, [F2] makes $q_h$ a classical $C^{2,\alpha }$ solution with uniformly Hölder-bounded forcing and values. Its principal coefficients are fixed, uniformly elliptic and Hölder by [F1]–[F2]. With Countable Choice from [F7], [F5] gives a uniform $C^{2,\alpha }$ bound for these quotients on a smaller ball. No derivative beyond the given order is used to apply this classical estimate.

Choose a closed coordinate cube $K$ about the point, contained in the smaller ball of step 1.1, and a sequence of positive admissible shifts $h_n\to 0$. The cube is a nonempty compact metric space by [F18]. For each fixed pair $(I,j)$ the finitely many fields $D^{\beta }{q}_{h_n}$, $\mathrm{\mid }\beta \mathrm{\mid }\le 2$, are uniformly bounded on $K$. The second-order fields have a common Hölder modulus. For each field of order at most one, join two points of $K$ by coordinate segments in $K$; [F13] on each segment and the common next-derivative bound give a common Lipschitz bound, hence equicontinuity. The derivative identifications at these known orders use [F14]. With the choice hypotheses supplied by [F7], apply [F6] successively to these finitely many real fields, retaining one subsequence on which $D^{\beta }{q}_{h_n}\to Z_{\beta }$ uniformly on $K$ for all $\mathrm{\mid }\beta \mathrm{\mid }\le 2$. Each $Z_{\beta }$ is continuous. The defining derivative of $v$ gives $Z_0=D_{j}v$ pointwise.

For $x$ in the interior of $K$, a nonzero sufficiently small signed $t$, a coordinate direction $i$, and $\mathrm{\mid }\beta \mathrm{\mid }\le 1$, apply [F13] to $D^{\beta }{q}_{h_n}$ along the segment from $x$ to $x+t{e}_i$. Since ${\mathrm{\partial }}_i{D}^{\beta }{q}_{h_n}=D^{\beta +e_i}{q}_{h_n}$ by [F14], this gives $\mid \frac{D^{\beta }{q}_{h_n}(x+t{e}_i)-D^{\beta }{q}_{h_n}(x)}{t}-Z_{\beta +e_i}(x)\mid \le \mathrm{\parallel }{D}^{\beta +e_i}{q}_{h_n}-Z_{\beta +e_i}{\mathrm{\parallel }}_{\mathrm{\infty };K}+{\sup }_{s\text{ between }0\text{ and }t}\mathrm{\mid }{Z}_{\beta +e_i}(x+s{e}_i)-Z_{\beta +e_i}(x)\mathrm{\mid }$. Let $n\to \mathrm{\infty }$ and then $t\to 0$. Uniform convergence and continuity give ${\mathrm{\partial }}_i{Z}_{\beta }=Z_{\beta +e_i}$. Thus $Z_0$ is $C^2$ on the interior of $K$, with the indicated derivatives. For $\mathrm{\mid }\beta \mathrm{\mid }=2$, passing to the limit in $\mathrm{\mid }{D}^{\beta }{q}_{h_n}(x)-D^{\beta }{q}_{h_n}(y)\mathrm{\mid }\le C\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }$ retains the same Hölder bound on $K$. Therefore $D_j{D}^{I}w=Z_0$ is $C^{2,\alpha }$ locally. This identifies the limits locally without using any boundary argument.

Every ordered derivative of $w$ of order $m-1$ equals some $D_j{D}^{I}w$ by [F14] at the already known $C^m$ regularity. By step 3.1 that field has every further ordered derivative through order two, continuously, with second derivatives Hölder. Any ordered word of length $m+1$ is two differentiations of such a length-$(m-1)$ field; hence all these words exist and are Hölder. The coordinate definition in [F9] now gives $w\in C^{m+1,\alpha }$ on a smaller neighborhood. There are only finitely many $(I,j)$ at each fixed order. Starting from $m=4$, repeat on nested balls whose radii decrease to a positive limit; each local gain is available on any prescribed compact smaller ball by a finite cover and shrinking the shifts. This establishes all finite orders on one neighborhood of each interior point and proves interior smoothness.

At the boundary, start with the supplied $C^{4,\alpha }$ regularity. For each integer $m\ge 4$ apply [F3], with its ZFC hypothesis supplied by [F7], to obtain the tangential gain, and then [F4], now legitimate in the smooth interior from step 4.1, to recover the remaining normal derivatives and obtain $C^{m+1,\alpha }$ up to a smaller patch. The suppliers are local at every point: applying them over a finite cover of any prescribed compact smaller half-patch gives the gain there. Choose nested half-patches with radii decreasing to a positive limiting radius, so every finite order is established on one common smaller half-patch. A finite boundary cover gives all-order continuous derivative extensions up to compact $S$.

Fix a flattened boundary point and translate it to $0$. Work in the common all-order half-patch of step 5.1. Choose $a>0$ small and $0<2\rho <a\mathrm{/}2$. For $x^{\mathrm{\prime }}=(x_1,x_2)$ with $\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }<a$, put $\mathrm{\ell }(x^{\mathrm{\prime }})=a-\sqrt{a^2-\mathrm{\mid }{x}^{\mathrm{\prime }}{\mathrm{\mid }}^2}$ and $u(x^{\mathrm{\prime }})=a+\sqrt{a^2-\mathrm{\mid }{x}^{\mathrm{\prime }}{\mathrm{\mid }}^2}$. These are the lower and upper graphs of the sphere of radius $a$ centered at $a{e}_3$. With the two-dimensional cutoff from [F15], put $\theta (x^{\mathrm{\prime }})=\eta (x^{\mathrm{\prime }}\mathrm{/}\rho )$ and $g=(1-\theta )\mathrm{\ell }$, and define $\mathrm{\Omega }=\{(x^{\mathrm{\prime }},x_3):\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }<a,g(x^{\mathrm{\prime }})<x_3<u(x^{\mathrm{\prime }})\}$. Here $0\le g\le \mathrm{\ell }<u$, so this is nonempty, bounded, open, and its interior lies in $x_3>0$. It is unchanged from the ball near $\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }=a$, because $\theta =0$ for $\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }\ge 2\rho$. For sufficiently small $r>0$ with $r<\rho$ and $r<a$, its intersection with $B_r$ is exactly $B_r^{+}$. Since $\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }\le a$ and $0\le x_3\le 2a$ on its closure, choosing $a$ sufficiently small places $\bar{\mathrm{\Omega }}$ inside the closed all-order half-patch.

Verify the smooth graph-domain condition required by [F11]. The sphere polynomial $Q(x)=\mathrm{\mid }{x}^{\mathrm{\prime }}{\mathrm{\mid }}^2+(x_3-a{)}^2-a^2$ has nonzero gradient at every zero. Choose any coordinate whose partial derivative there is nonzero, using a coordinate permutation as a rigid motion. By [F16] this polynomial is $C^1$, so [F17] gives a local $C^1$ graph in that coordinate. Its derivative is a quotient of polynomials evaluated on the graph with nonvanishing denominator. Iterating the derivative formula with [F16] and [F12] proves inductively that the graph is $C^k$ for every $k$, hence smooth. The nonzero coordinate derivative has fixed sign after shrinking, so the defining inequality $Q<0$ lies on exactly one side of the graph; a reflection if necessary writes it as $s<h(y)$ as required by [F11]. On $\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }<a$ this also proves smoothness of $\mathrm{\ell }$ and $u$, since ${\mathrm{\partial }}_{3}Q=2(x_3-a)\mathrm{\ne }0$ on either graph there. The modified cap $g=(1-\theta )\mathrm{\ell }$ is smooth by [F15]–[F16] and has the same one-sided graph property, again after reflecting the last coordinate. Near the seam $\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }=2\rho$ it is a smooth graph, and near the equator the domain is exactly the original ball. These neighborhoods cover the boundary; thus $\mathrm{\Omega }$ is a bounded smooth graph domain with the precise local rigid-motion and one-sided properties of [F11].

Restrict $w$ to $\mathrm{\Omega }$ and set $u(x,\tau )=w(x)$ for $\tau \in (-1,1)$. By steps 4.1 and 5.1 this function is smooth on $\mathrm{\Omega }\times (-1,1)$ and all mixed derivatives extend continuously to $\bar{\mathrm{\Omega }}\times (-1,1)$; positive parameter derivatives vanish. The domain is admissible by step 7.1. Apply [F11] with $V={\mathbb{R}}^3$. Evaluating its extension at $\tau =0$ gives a single smooth extension of $w$ agreeing with it on the closed local half-neighborhood from step 6.1. Hence [F9] gives smoothness up to $S$ in the required sense. Composing with the smooth flow by [F12] makes $F_w$ smooth. Its differential is still injective, its image topology is unchanged, and the positive definite induced metric is the one assumed in finite-regularity admissibility. By [F10], transport the smooth structure of $\mathrm{\Sigma }$ through this embedding to its image. Finally [F8] and $w{\mathrm{\mid }}_S=0$ give $F_w{\mathrm{\mid }}_S={\mathrm{i}\mathrm{d}}_S$, so its boundary is $S$. No new graph, weighted end estimate or parameter regularity is inferred from this smoothness argument.