Suppose I have got two bounded Lipschitz domains $A$ and $B$, both subsets of $\mathbb{R}^n$. Suppose I have a Lipschitz diffeomorphism $F\colon \bar A \to \bar B$. So $F$ is a Lipschitz function which is invertible and the inverse is also Lipschitz.
Does it follows that $F$ when restricted to the boundary $\partial A$, is also Lipschitz diffeomorphic to $\partial B$? I.e. is $F|_{\partial A}\colon \partial A \to \partial B$ a Lipschitz diffeomorphism? If not, what is the minimum regularity I need to ensure the regularity of the diffeomorphism is also kept for the restriction? Most results I have seen are for smooth diffeomorphisms, so I can't use those.
We can consider the Jacobian determinant of $F$, call it $J:=\mathrm{det}(\nabla F)$. Is it true that the Jacobian determinant of $F|_{\partial A}$ is just $J|_{\partial A}$?
If anyone has a reference to some source where properties of Lipschitz diffeos are discussed in length I would be grateful.