Thank you for such an elaborate answer, please point me towards some good sources where I can learn more about extention operator you mentioned.

@LeoMoos Please let me know the definition of zig zag function you are considering.

@LeoMoos Thanks for the comment, but the example you mentioned does not have a uniform bound on Holder norm. In fact the Holder norm of $f_n$ blows up as $n\rightarrow \infty$

nothing can be said when $A$ is just Holder? what about the second case, cant any holder continuous function be modulus of gradient of a $C^{1,\alpha}$ function?

I saw thhe problem bit differently, if the solution $\Phi$ is such that $D\Phi$ only need to preserve distances, then $D\Phi(x)= U(x) B (x) U(x)^{-1} $, for some distance preserving matric $U(x)$. where $B(x)= |det (A(x))|^{1/N+2} A(x)$. Now that we have more flexibility in solving the problem by choosing any $U$, does it not make the problem easier? (though I still not know the answer)

thank you for your answer, since I am not very familiar with differential forms, if yiu could clarify what "integrability condition" on $B_{ij}$ you are referring to? moreover, is the claim true if instead of equality, we have the following condition :$ |D\Phi (x) \xi| = \frac {|A(x)\xi|}{|det(A(x))|^{1/N+2}}$? this condition is more relaxed than one in question, instead of matrices being equal we only need to preserve distanced from otigin.