On the bi-Sobolev planar homeomorphisms and their approximation

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.