On the bi-Sobolev planar homeomorphisms and their approximation

TL;DR

This paper characterizes planar bi-Sobolev homeomorphisms, proves key properties, and demonstrates that such homeomorphisms with integrable inverse can be approximated by smooth diffeomorphisms in the Sobolev space.

Contribution

It provides a self-contained analysis of bi-Sobolev homeomorphisms and confirms that they can be approximated by smooth maps, resolving an open conjecture for the case p=1.

Findings

Proved that $Du(x)=0$ almost everywhere where $J_u(x)=0$.

Established the equality of integrals of $|Du|$ and $|Du^{-1}|$.

Showed approximation of $W^{1,1}$ homeomorphisms by smooth diffeomorphisms.

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 \begin{equation} \int_\Omega |Du| = \int_\Delta |Du^{-1}|\,. \end{equation} 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.\par 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~\cite[Question~4]{arXiv:1009.0286}) for the case $p=1$.