Regularity and non-degeneracy of ${\Phi }^{\ast }[I]$ 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 ${\Phi }^{\ast }[I]$ 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 $({\Phi }^{\ast }[I](P)-I)\nu \ne 0$ and the assumption that ${\Phi }^{\ast }[I]$ 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.

8 pages