Bi-Lipschitz NURBS maps preserve H1 and energy norms

First let $v\in C^{\mathrm{\infty }}(\mathrm{\Omega })\cap H^1(\mathrm{\Omega })$. On each open knot cell the differentiability expansions of $F$ and $v$ give ${\mathrm{\nabla }}_{\xi }(v\circ F)=D{F}^{\mathsf{T}}({\mathrm{\nabla }}_{x}v)\circ F$. Fix a test function compactly supported in $\widehat{\mathrm{\Omega }}$. Its support stays away from the exterior patch boundary. On every part of a cell where the test function is nonzero, its image under $F$ lies in a compact subset of $\mathrm{\Omega }$, so $v\circ F$ and its first derivatives are bounded there; near exterior faces the test function vanishes. Apply the divergence theorem [F6] to $(v\circ F)\varphi e_j$ cell by cell. The values of $v\circ F$ and $\varphi$ agree on either side of every shared face, so opposite normal fluxes cancel, and exterior terms vanish. This proves the weak-derivative identity without requiring $v$ to extend to $\mathrm{\partial }\mathrm{\Omega }$. Cellwise change of variables [F2], summed over the finite partition using finite additivity of the integral, yields $\mathrm{\parallel }v\circ F{\mathrm{\parallel }}_{L^2(\widehat{\mathrm{\Omega }})}^2\le j_{-}^{-1}\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2,\mathrm{\parallel }{\mathrm{\nabla }}_{\xi }(v\circ F){\mathrm{\parallel }}_{L^2(\widehat{\mathrm{\Omega }})}^2\le L^2{j}_{-}^{-1}\mathrm{\parallel }{\mathrm{\nabla }}_{x}v{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2\mathrm{.}$ Thus $\mathrm{\parallel }v\circ F{\mathrm{\parallel }}_{H^1(\widehat{\mathrm{\Omega }})}\le j_{-}^{-1\mathrm{/}2}\sqrt{1+L^2}\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1(\mathrm{\Omega })}$. The same argument applied to $F^{-1}$, whose Jacobian is $J_F^{-1}$ and derivative is bounded by $L$, gives the reverse estimate.

For arbitrary $u\in H^1(\mathrm{\Omega })$, choose $v_n\in C^{\mathrm{\infty }}(\mathrm{\Omega })\cap H^1(\mathrm{\Omega })$ with $v_n\to u$ in $H^1$ by [F1]. Step 1.1 applied to $v_n-v_m$ makes $v_n\circ F$ Cauchy in $H^1(\widehat{\mathrm{\Omega }})$; completeness [F4] gives a limit $\widehat{u}\in H^1(\widehat{\mathrm{\Omega }})$. The same change-of-variables estimate in step 1.1 implies $v_n\circ F\to u\circ F$ in $L^2(\widehat{\mathrm{\Omega }})$, so this limit is the composition class. The gradient formula follows by taking the $L^2$ limit of the step 1.1 gradients: the bound on $DF$ and the change-of-variables estimate imply $D{F}^{\mathsf{T}}(\mathrm{\nabla }{v}_n)\circ F\to D{F}^{\mathsf{T}}(\mathrm{\nabla }u)\circ F$ in $L^2$. Repeating the argument with $F^{-1}$ proves the reverse estimate and inverse pullback.

Suppose $u\in H_0^1(\mathrm{\Omega })$ and choose ${\phi }_n\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ converging to $u$ in $H^1$ by [F4]. Each ${\phi }_n\circ F$ belongs to $H^1(\widehat{\mathrm{\Omega }})$ by step 1.1 and has compact support in $\widehat{\mathrm{\Omega }}$, because $F^{-1}$ maps compact subsets of $\mathrm{\Omega }$ to compact subsets of $\widehat{\mathrm{\Omega }}$. Applying [F5] gives ${\phi }_n\circ F\in H_0^1(\widehat{\mathrm{\Omega }})$. The forward estimate in step 2.1 gives convergence in $H^1$, and closedness in [F4] implies $u\circ F\in H_0^1(\widehat{\mathrm{\Omega }})$. Applying the same reasoning to $F^{-1}$ proves the converse. Finally, the scalar energy identity follows by substituting ${\mathrm{\nabla }}_{x}u\circ F=D{F}^{-\mathsf{T}}{\mathrm{\nabla }}_{\xi }\widehat{u}$ and $dx=J_{F}d\xi$ into the defining integral. For a vector field with row-gradient convention, ${\mathrm{\nabla }}_{x}v=({\mathrm{\nabla }}_{\xi }\widehat{v})D{F}^{-1}$; hence ${\epsilon }_x(v)\circ F=\mathrm{sym}(({\mathrm{\nabla }}_{\xi }\widehat{v})D{F}^{-1})$ and ${\mathrm{div}}_{x}v\circ F=\mathrm{tr}(({\mathrm{\nabla }}_{\xi }\widehat{v})D{F}^{-1})$, which gives the stated elasticity transformation.