On the bi-Sobolev planar homeomorphisms and their approximation. (May 2017) - Pratelli, Aldo
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 : Ω → Δ, one has D u ( x ) = 0 for almost every point x for which J u ( x ) = 0 . As a consequence, one can prove that (8) ∫ Ω | D u | = ∫ Δ | D u − 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 Iwaniec et al. (2011), Question 4) for the case p = 1 .