A boundary-fixing diffeomorphism preserves the anisotropic DN map

The pushforward coefficient is admissible. Since $F$ and $F^{-1}$ are smooth on the compact closure, $DF$, $D{F}^{-1}$, $J_F$, and $J_F^{-1}$ are bounded, with $J_F$ positive. The formula in the statement is smooth. For each $\xi \in {\mathbb{R}}^n$ and $x=F(y)$, ${\xi }^T(F_{\ast }\gamma )(x)\xi =(DF(y{)}^T\xi {)}^T\gamma (y)(DF(y{)}^T\xi )\mathrm{/}{J}_F(y)$; the ellipticity of $\gamma$ and the uniform upper and lower singular-value bounds for $DF$ and $J_F$ give positive finite uniform ellipticity constants for $F_{\ast }\gamma$.

In the Riemannian setting of the statement, the chain rule [F3] gives $d(u\circ \mathrm{\Phi })=D{\mathrm{\Phi }}^T(du\circ \mathrm{\Phi })$ in a coordinate chart, while [F8] gives $({\mathrm{\Phi }}^{\ast }g)=D{\mathrm{\Phi }}^T(g\circ \mathrm{\Phi })D\mathrm{\Phi }$. Thus $({\mathrm{\Phi }}^{\ast }g{)}^{-1}=D{\mathrm{\Phi }}^{-1}(g^{-1}\circ \mathrm{\Phi })D{\mathrm{\Phi }}^{-T}$, and multiplication shows $⟨d(u\circ \mathrm{\Phi }),d(v\circ \mathrm{\Phi }){⟩}_{{\mathrm{\Phi }}^{\ast }g}=(⟨du,dv{⟩}_g)\circ \mathrm{\Phi }$. The determinant formula [F9] gives $d{V}_{{\mathrm{\Phi }}^{\ast }g}=\mathrm{\mid }\det D\mathrm{\Phi }\mathrm{\mid }(d{V}_g\circ \mathrm{\Phi })$, with the absolute Jacobian and no orientation choice.

Composition by $F^{-1}$ and by $F$ is bounded on $H^1(\mathrm{\Omega })$. For ambient smooth $u$, the chain rule [F3] and change of variables [F4], whose choice hypothesis follows from [A1] by [F11], yield ${\int }_{\mathrm{\Omega }}(\mathrm{\mid }u\circ F^{-1}{\mathrm{\mid }}^2+\mathrm{\mid }\mathrm{\nabla }(u\circ F^{-1}){\mathrm{\mid }}^2)dx\le C_F{\int }_{\mathrm{\Omega }}(\mathrm{\mid }u{\mathrm{\mid }}^2+\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2)dy$, because $D{F}^{\pm 1}$ and $J_F^{\pm 1}$ are bounded; the nonnegative change-of-variables formula applies to the displayed squared terms. By [F5], smooth functions are dense in $H^1$; extending this estimate to limits defines the bounded compositions for every $H^1$ class. For smooth approximants $u_k$, the pointwise boundary identity $(u_k\circ F^{-1}){\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=u_k{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}$ holds since $F^{-1}$ fixes $\mathrm{\partial }\mathrm{\Omega }$. Continuity of the trace [F6] and convergence of both compositions in $H^1$ therefore give $T(u\circ F^{-1})=Tu$. The same argument with $F$ in place of $F^{-1}$ gives $T(u\circ F)=Tu$. Since $\ker T=H_0^1(\mathrm{\Omega })$ by [F6], both trace equalities imply $[u\circ F^{-1}]=[u\circ F]=[u]$ in the quotient. Finally, if $\varphi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, then $\varphi \circ F^{-1}$ and $\varphi \circ F$ are smooth and compactly supported in $\mathrm{\Omega }$, because $F$ maps the interior to itself and maps compact sets to compact sets. Applying bounded composition to sequences converging in [F7] proves that both compositions preserve $H_0^1(\mathrm{\Omega })$. The weak-gradient chain rule and the energy change-of-variables identity also extend to all $H^1$ classes. Given $u\in H^1(\mathrm{\Omega })$, choose ambient smooth restrictions $u_k\to u$ in $H^1$ by [F5]. For each $k$, the classical chain rule gives $\mathrm{\nabla }(u_k\circ F^{-1})(F(y))=DF(y{)}^{-T}\mathrm{\nabla }{u}_k(y),\mathrm{\nabla }(u_k\circ F)(y)=DF(y{)}^T\mathrm{\nabla }{u}_k(F(y))\mathrm{.}$ The composition estimates just established give convergence of the left sides in $L^2$; the uniform bounds on $D{F}^{\pm 1}$ and the Jacobian, together with [F4], give convergence of the right sides in $L^2$. Passing to the limits identifies the weak gradients of $u\circ F^{-1}$ and $u\circ F$ by the same formulas. Applying these identities to $u,v\in H^1(\mathrm{\Omega })$ and using [F4] on the positive and negative parts of the integrable energy density yields ${\int }_{\mathrm{\Omega }}(F_{\ast }\gamma )(x)\mathrm{\nabla }(u\circ F^{-1})(x)\cdot \mathrm{\nabla }(v\circ F^{-1})(x)dx={\int }_{\mathrm{\Omega }}\gamma (y)\mathrm{\nabla }u(y)\cdot \mathrm{\nabla }v(y)dy\mathrm{.}$ Integrability follows from boundedness of the coefficients and Cauchy--Schwarz.

The pointwise density identity in step 1.2 makes the left density the pullback of $⟨du,dv{⟩}_{g}d{V}_g$. The boundary has zero Riemannian volume: in each boundary chart it is contained in a coordinate hyperplane, which has Lebesgue measure zero by Fubini, and the smooth volume density is locally bounded. Cover $M$ by finitely many source neighborhoods each lying in a chart and mapped by $\mathrm{\Phi }$ into another chart; such neighborhoods exist by continuity and compactness. Partition their interiors into finitely many disjoint measurable pieces subordinate to this cover. On each piece apply the Euclidean change-of-variables formula [F4], whose choice hypothesis follows from [A1] by [F11], to the positive and negative parts of the smooth integrand, extended by zero within the corresponding chart. Summing the finitely many equalities and ignoring the null boundary gives ${\int }_M⟨d(u\circ \mathrm{\Phi }),d(v\circ \mathrm{\Phi }){⟩}_{{\mathrm{\Phi }}^{\ast }g}d{V}_{{\mathrm{\Phi }}^{\ast }g}={\int }_M⟨du,dv{⟩}_{g}d{V}_g$.

Let $f\in {\mathcal{B}}_{\mathrm{\Omega }}$ and let $u=u_{\gamma ,f}$. Set $v=u\circ F^{-1}$. By step 2.1 its boundary class is $f$. For any $\phi \in H_0^1(\mathrm{\Omega })$, $\phi \circ F\in H_0^1(\mathrm{\Omega })$ by step 2.1. The energy identity of step 2.1, with second input $\phi \circ F$, gives ${\int }_{\mathrm{\Omega }}(F_{\ast }\gamma )(x)\mathrm{\nabla }v(x)\cdot \mathrm{\nabla }\phi (x)dx={\int }_{\mathrm{\Omega }}\gamma (y)\mathrm{\nabla }u(y)\cdot \mathrm{\nabla }(\phi \circ F)(y)dy=0$. The matrix factors cancel because $D{F}^{T}D{F}^{-T}=I$, and the Jacobian $J_F$ in $dx$ cancels the denominator in $F_{\ast }\gamma$. Thus $v$ is the weak solution with coefficient $F_{\ast }\gamma$ and boundary class $f$; uniqueness [F2] identifies $v=u_{F_{\ast }\gamma ,f}$.

Fix any $h\in {\mathcal{B}}_{\mathrm{\Omega }}$ and a representative $V_h\in H^1(\mathrm{\Omega })$. Since $F$ fixes the boundary, step 2.1 gives $V_h\circ F$ the same boundary class $h$. The energy identity of step 2.1, with second input $V_h\circ F$, and the solution identification in step 3.1 give $⟨{\mathrm{\Lambda }}_{F_{\ast }\gamma }f,h⟩={\int }_{\mathrm{\Omega }}(F_{\ast }\gamma )\mathrm{\nabla }(u\circ F^{-1})\cdot \mathrm{\nabla }{V}_{h}dx={\int }_{\mathrm{\Omega }}\gamma \mathrm{\nabla }u\cdot \mathrm{\nabla }(V_h\circ F)dy=⟨{\mathrm{\Lambda }}_{\gamma }f,h⟩$. This holds for every $f,h$, so the bounded bilinear maps are equal.

If $u$ is smooth and $g$-harmonic, then for any $\psi \in C_c^{\mathrm{\infty }}(M^{\circ })$, $\psi \circ {\mathrm{\Phi }}^{-1}$ is also smooth and compactly supported in $M^{\circ }$. Applying the identity of step 2.2 to $u$ and $\psi \circ {\mathrm{\Phi }}^{-1}$ yields ${\int }_M⟨d(u\circ \mathrm{\Phi }),d\psi {⟩}_{{\mathrm{\Phi }}^{\ast }g}d{V}_{{\mathrm{\Phi }}^{\ast }g}=0$, so $u\circ \mathrm{\Phi }$ is ${\mathrm{\Phi }}^{\ast }g$-harmonic. Because $\mathrm{\Phi }$ fixes $\mathrm{\Gamma }$ and is bijective, it maps $\mathrm{\partial }M\setminus \mathrm{\Gamma }$ to itself; therefore it leaves unchanged every boundary function supported in $\mathrm{\Gamma }$. Step 2.2 gives equality of the local measurements for every such pair. When the stated Dirichlet solutions are unique, these harmonicity and trace facts identify the solution for ${\mathrm{\Phi }}^{\ast }g$ with the pullback of the solution for $g$, so their local energy maps agree.