Flattening a smooth fixed boundary preserves the elliptic graph equation

Choose a smooth boundary chart with $S$ represented by $x_3=0$ and the reference interior by $x_3>0$. Write $y=\chi (x)$ for its smooth change from reference coordinates and $J_i{}^a=\mathrm{\partial }{x}^a\mathrm{/}\mathrm{\partial }{y}^i$. Applying the chain rule twice gives $w_{y^i{y}^j}=J_i{}^a{J}_j{}^b{w}_{x^a{x}^b}+({\mathrm{\partial }}_{y^i}{J}_j{}^b)w_{x^b}$. Thus the symmetric principal matrix of the expression in F1 transforms to ${\tilde{C}}^{ab}=C^{ij}{J}_i{}^a{J}_j{}^b$, while the terms involving second derivatives of the coordinate change enter the lower-order coefficient. Both coefficients remain smooth functions of $(x,z,p)$, where $z$ and $p$ denote height and slope. The stipulated local extensions handle the flat boundary, and $w{\mathrm{\mid }}_S=0$ gives zero height there.

In the flow coordinates the graph tangent vectors are $e_i={\mathrm{\partial }}_{x^i}+w_i{\mathrm{\partial }}_s$ and $Y={\mathrm{\partial }}_s$. Comparing their $x$ components shows that $Y$ is never tangent to any such graph. Since the induced metric is positive definite, its tangent space is the full orthogonal complement of the future unit normal $n$; consequently $a=-g(Y,n)\mathrm{\ne }0$. The normal and $a$ depend smoothly on the spacelike first jet. Around any fixed admissible jet shrink to a connected jet neighborhood on which $a$ has one sign $\epsilon \in \{-1,1\}$. For a height variation $v$, the variational vector is $vY=(av)n+v{Y}^{\mathrm{\top }}$. By F2 its only second derivatives of $v$ are $a{h}^{ij}{v}_{ij}$: expanding ${\mathrm{\Delta }}_h(av)$ leaves all other terms with at most one derivative of $v$, and the tangential term is $v{Y}^{\mathrm{\top }}(H)$. Differentiating the quasilinear expression in F1 with respect to the height gives principal coefficient $C^{ij}$, so $C^{ij}=a{h}^{ij}$. This comparison can be made on smooth test graphs realizing each prescribed finite jet; the finite-jet formula then gives the same identity at the stated $C^{4,\alpha }$ regularity. Multiplying both coefficients of the equation by $\epsilon$ gives $A^{ij}=\mathrm{\mid }a\mathrm{\mid }{h}^{ij}$ before coordinate change, hence a positive definite matrix also after the congruence in step 1.1. If $Y$ is future timelike, writing $Y=an+Y^{\mathrm{\top }}$ shows $a>0$ by the future orientation, so $\epsilon =+1$.

Choose a sufficiently small closed coordinate half-box and closed height-and-slope boxes around the boundary jet, wholly within the chart's spacelike-jet domain and the sign neighborhood of step 2.1. This is possible by openness relative to the boundary half-space. Their product is compact by F8. On its product with the Euclidean unit sphere, the quadratic form $(x,z,p,\xi )\mapsto A^{ij}(x,z,p){\xi }_i{\xi }_j$ is continuous and strictly positive. That product is closed and bounded and hence compact by F8; F7 gives a strictly positive minimum $\lambda$ and a finite maximum $\mathrm{\Lambda }$. By homogeneity, $\lambda \mathrm{\mid }\xi {\mathrm{\mid }}^2\le A^{ij}{\xi }_i{\xi }_j\le \mathrm{\Lambda }\mathrm{\mid }\xi {\mathrm{\mid }}^2$ for every $\xi$. This proves uniform ellipticity on the smaller compact graph-jet neighborhood. An interior coordinate box gives the same argument without the boundary half-space.

On a smaller flat patch, continuous boundary extensions of the derivatives allow tangential differentiation of the trace. Explicitly, the one-variable mean value identity from F9 on an interior tangential segment gives $w(x^{\mathrm{\prime }}+t{e}_j,r)-w(x^{\mathrm{\prime }},r)=t{\mathrm{\partial }}_{j}w(x^{\mathrm{\prime }}+\theta t{e}_j,r)$ for some $0<\theta <1$, where $j<3$ and $r>0$. Let $r$ decrease to zero. Continuity of ${\mathrm{\partial }}_{j}w$ near $(x^{\mathrm{\prime }},0)$ bounds the difference between the resulting boundary difference quotient and ${\mathrm{\partial }}_{j}w(x^{\mathrm{\prime }},0)$ by ${\sup }_{\mathrm{\mid }s\mathrm{\mid }\le \mathrm{\mid }t\mathrm{\mid },r\ge 0\text{ small}}\mathrm{\mid }{\mathrm{\partial }}_{j}w(x^{\mathrm{\prime }}+s{e}_j,r)-{\mathrm{\partial }}_{j}w(x^{\mathrm{\prime }},0)\mathrm{\mid }$, which tends to zero as the segment and its height shrink. Hence the tangential derivative of the trace is the trace of ${\mathrm{\partial }}_{j}w$. Repeat this argument with each already available purely tangential derivative. Since the initial trace is identically zero, all these positive-order tangential traces vanish. The repetition applies through every finite order for which the derivatives extend continuously to the boundary, in particular through order four under the stated hypotheses.