*Published Paper*

**Inserted:** 3 sep 2015

**Last Updated:** 1 sep 2017

**Journal:** Nonlinear Anal.

**Year:** 2015

**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 $u:\Omega\to \Delta$, one has $Du(x)=0$ for almost every point $x$ for which $J_u(x)=0$. As a consequence, one can prove that $\int_\Omega

Du

= \int_\Delta

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 $W^{1,1}$ 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 [IwaniecKovalevOnninen,Question~4]) for the case $p=1$.

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