$L^p$ Approximation of maps by diffeomorphisms

Abstract.

It is shown that if $d \geq 2,$ then every map $\phi: \Omega \subset{\bb R}^d \rightarrow{\bb R}^d$ of class $L^\infty$ can be approximated in the $L^p$-norm by a sequence of orientation-preserving diffeomorphims $\phi_n: \bar \Omega \rightarrow \phi_n(\bar \Omega).$ These conclusions hold provided that $\Omega \subset{\bb R}^d$ is open, bounded, and that $1 \leq p <+\infty.$ In addition, $\phi_n(\bar \Omega)$ is contained in the $1/n$-neighborhood of the convex hull of $\phi(\Omega).$ All these conclusions fail for $\Omega \subset{\bb R}.$ The main ingredients of the proof are the polar factorization of maps [4] and an approximation result for measure-preserving maps on the unit cube for which we provide a proof based on the concept of doubly stochastic measures (Corollary 1.1).