Regularity and non-degeneracy of $\Phi^*[I]$ implies regularity of the fixed boundary $\partial\Omega$

Abstract

For a diffeomorphism $\Phi$ of a domain $\overline{\Omega}$ onto itself, which is the identity on $\partial\Omega$, we prove that local regularity of the push-forward $\Phi^*[I]$ implies local regularity of $\partial\Omega$, provided a certain non-degeneracy condition is satisfied. To be precise, if $\nu$ is a normal to $\partial\Omega$ at a point $P$, then the condition $(\Phi^*[I](P)-I)\nu \neq 0$ and the assumption that $\Phi^*[I]$ is of class $C^{k+1,\alpha}$ near $P$ imply that $\partial\Omega$ is also of class $C^{k+1,\alpha}$ near $P$. This result naturally complements recent regularity results for non-scattering inhomogeneities.