$L^p$ Approximation of maps by diffeomorphisms

It is shown that if then every map of class can be approximated in the -norm by a sequence of orientation-preserving diffeomorphims These conclusions hold provided that is open, bounded, and that In addition, is contained in the -neighborhood of the convex hull of All these conclusions fail for 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). Received: 25 My 2001 / Accepted: 25 October 2001 / Published online: 29 April 2002 RID="*" ID="*" en détachement de l'Université Paris 6, France The second author gratefully acknowledges the support of National Science Foundation grants DMS-99-70520, and DMS-00-74037