Approximation of Piecewise Affine Homeomorphisms by Diffeomorphisms

Mora-Corral, Carlos; Pratelli, Aldo

doi:10.1007/s12220-012-9378-1

Cited by 28 publications

(36 citation statements)

References 20 publications

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“…In this section we outline the basic plan of our proof, to underline the main steps and help the reading of the construction. We remind the reader that our aim is to find an approximation done with piecewise affine homeomorphisms, and then the existence of an approximation with smooth diffeomorphisms will eventually immediately follow applying the result of [27].…”

Section: Introductionmentioning

confidence: 99%

Diffeomorphic approximation of $W^{1,1}$ planar Sobolev homeomorphisms

Hencl

Pratelli

2018

J. Eur. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

Diffeomorphic approximation of $W^{1,1}$ planar Sobolev homeomorphisms

Hencl

Pratelli

2018

J. Eur. Math. Soc.

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“…Proof of Theorem A. First of all we remark that, as usual, it is enough to show the result for the case of piecewise affine approximating homeomorphisms, because then the case of the diffeomorphisms follows automatically thanks to [12]. As a consequence, from now on we look for approximating homeomorphisms which are piecewise affine.…”

Section: Approximation Of Bi-sobolev Homeomorphismsmentioning

confidence: 94%

“…Now, we pass to consider Ω G . First of all, putting together (11), (12) and (13) we already know that…”

Section: Approximation Of Bi-sobolev Homeomorphismsmentioning

confidence: 99%

On the bi-Sobolev planar homeomorphisms and their approximation

Pratelli

2017

Nonlinear Analysis: Theory, Methods & Applications

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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 → ∆, one has Du(x) = 0 for almost every point x for which Ju(x) = 0. As a consequence, one can prove thatNotice 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) un in such a way that un converges to u in W 1,1 and, at the same time, u −1 n converges to u −1 in W 1,1 . This positively answers an open conjecture (see for instance [11, Question 4]) for the case p = 1.

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“…Our proof is constructive, thus long, but it relies essentially on three known facts: the Lebesgue differentiation Theorem for L 1 -maps in R d , the Jordan curve theorem and a planar bi-Lipschitz extension theorem for homeomorphic images of squares proved in [11]. As already mentioned in the introduction, all our effort will be to get a piecewise affine approximation of u, since then the smooth extension readily follows by the following recent result from [28].…”

Section: Scheme Of the Proof And Plan Of The Papermentioning

confidence: 97%

“…Thanks to a result by Mora-Corral and the second author [28] (see Theorem 2.1 below), the problem of finding smooth approximations can be actually reduced to find countably piecewise affine ones -i.e. affine on the elements of a locally finite triangulation of Ω, see Definition 3.2.…”

Section: )mentioning

confidence: 99%