Uniform elliptic extension and flattening before the boundary gradient estimate

If $\mathcal{K}$ is empty, use the identity matrix. Otherwise choose a bounded open neighborhood ${\mathcal{O}}_0$ of $\mathcal{K}$ with compact closure contained in $\mathcal{O}$: cover the compact set by finitely many small coordinate balls whose closed larger balls lie in $\mathcal{O}$, and take their union. Apply [F3] with $K=\mathcal{K}$ and $O={\mathcal{O}}_0$. Define $\tilde{A}=I+\zeta (A-I)$ on $\mathcal{O}$ and $\tilde{A}=I$ outside $\mathcal{O}$. Since the closed support of $\zeta$ is contained in ${\mathcal{O}}_0$, this formula equals $I$ on a neighborhood of every point outside that support; it therefore defines a smooth matrix on the whole coordinate jet space. The original $A$ is evaluated only inside its supplied domain, and $\tilde{A}=A$ on $\mathcal{K}$. Apply this construction in each of the finitely many patches.

On the compact closure of ${\mathcal{O}}_0$, [F1] gives $\lambda I\le A\le \mathrm{\Lambda }I$ for common $0<\lambda \le \mathrm{\Lambda }$. Since $\tilde{A}=\zeta A+(1-\zeta )I$, its quadratic form lies between $\min (\lambda ,1)\mathrm{\mid }\xi {\mathrm{\mid }}^2$ and $\max (\mathrm{\Lambda },1)\mathrm{\mid }\xi {\mathrm{\mid }}^2$. Smooth coefficient and cutoff derivatives have finite bounds on that compact set by [F4]; outside the cutoff support the extension is constant. Applying the product rule [F6] to each scalar coordinate restriction thus gives common joint $C^1$ bounds on any containing full finite height/slope box. Finite maxima over the background patches give common constants. Likewise the original smooth $b$ is bounded on $\mathcal{K}$, so $f=-b(x,u,Du)$ is bounded at every realized jet, without differentiating $Du$.

Under $y=\mathrm{\Phi }(x)$ write $q=D_y\tilde{u}$ and $p=D\mathrm{\Phi }(x{)}^{T}q$. Applying [F5] twice and the product rule [F6] gives $u_{ij}={\mathrm{\Phi }}_i^a{\mathrm{\Phi }}_j^b{\tilde{u}}_{ab}+{\mathrm{\Phi }}_{ij}^a{q}_a$. Thus the transformed principal matrix is ${\widehat{A}}^{ab}={\mathrm{\Phi }}_i^a{\tilde{A}}^{ij}(x,z,D{\mathrm{\Phi }}^{T}q){\mathrm{\Phi }}_j^b$. The bounds on $D\mathrm{\Phi }$ and its inverse preserve its two ellipticity constants up to common factors. By [F5] and [F6], the joint first derivatives of this expression use at most $D^2\mathrm{\Phi }$ and first joint derivatives of $\tilde{A}$, hence have common bounds. The transformed containing box may be enlarged to contain the bounded transformed heights and slopes. For an actual solution the right side is $\widehat{f}=-b(x,u,Du)-A^{ij}(x,u,Du){\mathrm{\Phi }}_{ij}^a{q}_a$, evaluated at $x={\mathrm{\Phi }}^{-1}(y)$. Every factor is bounded by the supplied chart bounds, gradient bound and step 2.1. Therefore $\widehat{f}$ is uniformly bounded as well; no derivative of the evaluated gradient appears.

A base-coordinate change leaves the scalar height value unchanged: $\tilde{u}(y)=u({\mathrm{\Phi }}^{-1}(y))$. Restricting this identity to the actual flattened boundary gives the trace formula in the Statement. It preserves an originally zero trace and transports any other trace rather than replacing it by zero. On interior patches step 2.1 applies without a trace assumption. These observations and step 3.1 establish every claimed reduction.