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- W4299849603 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 $u:Omegato Delta$, 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 $p=1$." @default.
- W4299849603 created "2022-10-03" @default.
- W4299849603 creator A5071174138 @default.
- W4299849603 date "2015-09-03" @default.
- W4299849603 modified "2023-09-28" @default.
- W4299849603 title "On the bi-Sobolev planar homeomorphisms and their approximation" @default.
- W4299849603 doi "https://doi.org/10.48550/arxiv.1509.01045" @default.
- W4299849603 hasPublicationYear "2015" @default.
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