L Approximation of maps by diffeomorphisms

It is shown that if d ≥ 2, then every map φ : Ω ⊂ R → Rd of class L∞ can be approximated in the Lp-norm by a sequence of orientationpreserving diffeomorphims φn : Ω̄ → φn(Ω̄). These conclusions hold provided that Ω ⊂ Rd is open, bounded, and that 1 ≤ p < +∞. In addition, φn(Ω̄) is contained in the 1/n-neighborhood of the convex hull of φ(Ω). All these conclusions fail for Ω ⊂ 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 new proof based on the concept of doubly stochastic measures (corollary 1.5). ∗WG gratefully acknowledges the support of National Science Foundation grants DMS99-70520, and DMS-00-74037.