On the bi-Sobolev planar homeomorphisms and their approximation

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 : Omega -> Delta, one has Du(x)= 0 for almost every point x for which J(u)(x) = 0. As a consequence, one can prove that integral(Omega) vertical bar Du vertical bar = integral(Delta) vertical bar Du(-1)vertical bar. (8) 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,W-1 homeomorphism u with W-1,W-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,W-1 and, at the same time, u(n)(-1) converges to u(-1) in W-1,W-1. This positively answers an open conjecture (see for instance Iwaniec et al. (2011), Question 4) for the case p = 1. (C) 2016 Published by Elsevier Ltd.