Assume $\Omega$ is a smoothly bounded open domain in $ \mathbb R^n$ and $B$ is an open ball of equal volume. Let $\Phi:\partial\Omega\to \partial B$ be a diffeomorphism. Is it true that there exists a volume preserving diffeomorphism $\tilde{\Phi}:\overline{\Omega}\to\overline{B}$ such that $\tilde{\Phi}\vert_{\partial\Omega}=\Phi$?
Volume Preserving Diffeomorphism – Extension Analysis
dg.differential-geometry
Best Answer
Let $\Omega$ be a submanifold of $\mathbb R^n$ with boundary $\partial\Omega$ diffeomorphic to $\partial B$. As far as I know, it is not obvious that $\overline\Omega$ should be diffeomorphic to $\overline B$, and my guess would be that this is actually false in all generality. From what I understand, though, it is true if you assume $\Omega$ is contractible and $n\geq5$, see this question.
So there might be a topological obstruction. Once you get past this (for instance if you already assume $\overline\Omega$ is diffeomorphic to $\overline B$), then it reduces to the same question with $\Omega=B$. Now there might be another topological obstruction, which is that the diffeomorphism of the boundary may not extend to a diffeomorphism of the whole ball, see this answer.
Again, if you already know that $\overline\Omega$ is diffeomorphic to $\overline B$, then these obstructions cannot happen, more or less by definition. Then your question becomes: let $\omega_1,\omega_2$ be two smooth volume forms on $B$ with the same global volume, is there a diffeomorphism sending one to the other? This is a Moser-type theorem, and the version for manifolds with boundary is proved in Formes-volumes sur les variétés à bord (volume forms on manifolds with boundary) by A. Banyaga (1974).