On the bi-Sobolev planar homeomorphisms and their approximation

Abstract

Listen

Translate article

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 Du(x)=0 for almost every point x for which Ju(x)=0. As a consequence, one can prove that (8)∫Ω|Du|=∫Δ|Du−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 W1,1 homeomorphism u with W1,1 inverse can be approximated with smooth diffeomorphisms (or piecewise affine homeomorphisms) un in such a way that un converges to u in W1,1 and, at the same time, un−1 converges to u−1 in W1,1. This positively answers an open conjecture (see for instance Iwaniec et al. (2011), Question 4) for the case p=1.