On the bi-Sobolev planar homeomorphisms and their approximation. (May 2017)

• Record Type: Journal Article
• Authors: Pratelli, Aldo
• Is Part Of: Nonlinear analysis. Volume 154(2017)
• Page Start: 258
• Page End: 268
• Publication Date: 2017-05
• Subjects: Bi-Sobolev homeomorphisms -- Smooth approximation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.07.006
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: The first goal of this paper is to give a short description of the planar bi-Sobolev homeomorphisms, providing simple and self-contained proofs for some already known properties. In particular, for any such homeomorphism u : Ω → Δ, one has D u ( x ) = 0 for almost every point x for which J u ( x ) = 0 . As a consequence, one can prove that (8) ∫ Ω | D u | = ∫ Δ | D u − 1 | . Notice that this estimate holds trivially if one is allowed to use the change of variables formula, but this is not always the case for a bi-Sobolev homeomorphism. As a corollary of our construction, we will show that any W 1, 1 homeomorphism u with W 1, 1 inverse can be approximated with smooth diffeomorphisms (or piecewise affine homeomorphisms) u n in such a way that u n converges to u in W 1, 1 and, at the same time, u n − 1 converges to u − 1 in W 1, 1 . This positively answers an open conjecture (see for instance Iwaniec et al. (2011), Question 4) for the case p = 1 .