Weak divergence-form equations are invariant under $C^2$ boundary charts

Local pullback and gradient. Let $G⋐V\cap H$ be open and choose $G^{\mathrm{\prime }}⋐W\cap \mathrm{\Omega }$ containing $\psi (\bar{G})$. By [F3], smooth approximations $u_m\to u$ in $H^1(G^{\mathrm{\prime }})$ exist. Change of variables and the compact chart bounds give $\mathrm{\parallel }(u_m-u_{\mathrm{\ell }})\circ \psi {\mathrm{\parallel }}_{H^1(G)}\le C_G\mathrm{\parallel }{u}_m-u_{\mathrm{\ell }}{\mathrm{\parallel }}_{H^1(G^{\mathrm{\prime }})}$. Passing to the $H^1$ limit establishes $\widehat{u}\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(V\cap H)$ and $D\widehat{u}=D{\psi }^T(Du\circ \psi )$ almost everywhere. If these derivative and Jacobian bounds are uniform on the whole matched patch, the identical integral estimate establishes global $H^1$ membership there. Local boundedness alone gives only the local conclusion.

Admissible transformed tests. For $\widehat{v}\in C_c^{\mathrm{\infty }}(V\cap H)$, [F5] makes $v=\widehat{v}\circ \mathrm{\Phi }$ an admissible compactly supported $H^1$ test. Approximate $v$ by smooth tests on a fixed compact patch; the coefficient bounds, $f\in L^2$ on that patch and Cauchy--Schwarz pass the weak identity to $v$. Thus no preservation of smooth test functions by the chart is required.

Coefficient matching. Use the standard Jacobian convention $M_{ip}={\mathrm{\partial }}_p{\mathrm{\Phi }}_i$ at $x=\psi (y)$ and $N_{pi}={\mathrm{\partial }}_i{\psi }_p$ at $y$, so $MN=NM=I$. The displayed component formula is $\tilde{a}=\mathrm{\mid }\det N\mathrm{\mid }Ma{M}^T$, and $Du(x)=M^{T}D\widehat{u}(y)$, $Dv(x)=M^{T}D\widehat{v}(y)$. Substituting these two gradients and $dx=\mathrm{\mid }\det N\mathrm{\mid }dy$ gives exactly $\int {\tilde{a}}^{ij}{D}_j\widehat{u}\overline{D_i\widehat{v}}dy=\int a^{pq}{D}_{q}u\overline{D_{p}v}dx$. The same substitution gives $\tilde{b}=\mathrm{\mid }\det N\mathrm{\mid }Mb$, $\tilde{c}=\mathrm{\mid }\det N\mathrm{\mid }c$ and $\widehat{f}=\mathrm{\mid }\det N\mathrm{\mid }(f\circ \psi )$. All integrals may be restricted to the matched test-support patches.

Global membership after localization. For $\zeta \in C_c^{\mathrm{\infty }}(W)$, the product $\zeta u$ belongs to $H^1(W\cap \mathrm{\Omega })$ and is supported in a compact ambient patch. On this patch $D\mathrm{\Phi },D\psi$ and the Jacobians have uniform bounds. The local chain rule of step 1.1 and change of variables give $\mathrm{\parallel }\widehat{\zeta u}{\mathrm{\parallel }}_{H^1(V\cap H)}\le C(\zeta ,\mathrm{\Phi })\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1(\mathrm{\Omega })}$ by integrating the weak gradient formula over the whole half-patch. The formula vanishes outside the image of the cutoff support. Its distributional equation follows by the product rule; with $a\in W^{1,\mathrm{\infty }}$ the cutoff commutators expand to $L^2$ terms whenever the localized forcing is $L^2$.

Zero-trace transfer on an aligned boundary patch. For a boundary chart whose ambient patch $W_0$ satisfies $\mathrm{\Phi }(W_0\cap \mathrm{\Omega })=V\cap H$, take an ambient cutoff $\zeta$ supported inside $W_0$ and $u\in H_0^1(\mathrm{\Omega })$. Approximate $u$ by smooth compactly supported functions in $\mathrm{\Omega }$, multiply by $\zeta$, and pull back. These pullbacks have compact support inside the open half-patch and lie in $H_0^1(V\cap H)$ by smooth approximation; uniform compact ambient chart bounds give their $H^1$ convergence. Hence the localized pullback has zero trace. Alignment is essential: restricting a compactly supported function across an unrelated interior plane does not preserve zero trace.

Higher-order localized pullback. On compact interior subsets, the smooth-approximation proof of C^k boundary flattening preserves local W^{k,p} gives $D^{\alpha }(z\circ \psi )={\sum }_{\mathrm{\mid }\beta \mathrm{\mid }\le \mathrm{\mid }\alpha \mathrm{\mid }}(D^{\beta }z)\circ \psi P_{\alpha \beta }(D\psi ,\dots ,D^m\psi )$ for $\mathrm{\mid }\alpha \mathrm{\mid }\le m$; order zero is the original class. The polynomials have uniform bounds on the compact ambient cutoff patch, including its flat boundary. Change of variables consequently bounds each field in $L^2$ on the entire half-patch, not just on its compact interior subsets. The local test identities identify these fields as the global weak derivatives; the cutoff vanishes near artificial edges, so no extra derivative is introduced by zero extension there. For $W^{m,\mathrm{\infty }}$ input, apply the finite-exponent formula on bounded interior subsets and observe directly that all its fields have a common essential bound on the half-patch. This proves the two claimed higher-order bounds. The multiplier and support facts are those of The cutoff difference-quotient commutator estimate.

Transformed equation. Steps 1.2 and 2.1 transform the actual weak identity for $u$ into $\int ({\tilde{a}}^{ij}{D}_j\widehat{u}\overline{D_i\widehat{v}}+{\tilde{b}}^i{D}_i\widehat{u}\bar{\widehat{v}}+\tilde{c}\widehat{u}\bar{\widehat{v}})=\int \widehat{f}\bar{\widehat{v}}$ for each smooth compactly supported transformed test. The transformed datum is locally $L^2$ by change of variables on compact patches. On every bounded $G⋐V\cap H$, step 1.1 gives $\widehat{u}\in H^1(G)$ and the chart bounds make all coefficients bounded. For $M=D\mathrm{\Phi }\circ \psi$ and $J=\mathrm{\mid }\det D\psi \mathrm{\mid }$, ellipticity gives $\mathrm{Re}({\tilde{a}}^{ij}{\xi }_j\overline{{\xi }_i})\ge J\theta \mathrm{\mid }{M}^T\xi {\mathrm{\mid }}^2\ge \theta ({\inf }_{G}J)({\sup }_G\mathrm{\parallel }D\psi \mathrm{\parallel }{)}^{-2}\mathrm{\mid }\xi {\mathrm{\mid }}^2$; this positive constant establishes the operator hypotheses on $G$. Thus the cited local-solution definition applies to each restriction. The unrestricted transformed identity requires only $H_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ membership and local coefficient bounds.

Conclusion. The component formulas and local weak equation are established by steps 1.1--3.2. Uniform chart bounds give global $H^1$ transfer, and step 2.3 proves zero-trace transfer for the aligned boundary patches used in Dirichlet estimates. The ambient cutoff supplies the uniform bounds for the global conclusion in step 2.2, and boundary alignment is a hypothesis of the Statement.