Bi-Sobolev homeomorphism with zero Jacobian almost everywhere

D'Onofrio, Luigi; Hencl, Stanislav; Schiattarella, Roberta

doi:10.1007/s00526-013-0669-6

Cited by 22 publications

(12 citation statements)

References 13 publications

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“…Note also that, if 1 ≤ p < n, then there are very pathological examples of Sobolev homeomorphisms with the Jacobian equal zero a.e. The first such example was constructed by Hencl [20], see also [5,8].…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Chain rule. The main result of [8] (see also [33]) provides an example of a surjective homeomorphism f : (0, 1) n → (0, 1) n , n ≥ 3, such that f ∈ W 1,1 , f −1 ∈ W 1,1 and J f = 0 a.e., J f −1 = 0 a.e. Note that f −1 • f = Id, but the chain rule…”

Section: Preliminariesmentioning

confidence: 99%

“…Denote the points of the cube (0, 1) n by (x, y) ∈ (0, 1) n−ℓ × (0, 1) ℓ = (0, 1) n and define f x (y) = f (x, y), f i,x (y) = f i (x, y). Then for almost all x ∈ (0, 1) n−ℓ we have f x , f i,x ∈ W 1,p (0, 1) ℓ and there is a subsequence f i j such that for almost all x ∈ (0, 1) n−ℓ (8) lim…”

Section: Lemma 18 (The Lebesgue Differentiation Theoremmentioning

confidence: 99%

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Jacobians of $W^{1,p}$ homeomorphisms, case $p=[n/2]$

Goldstein¹,

Hajłasz²

2018

Preprint

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We investigate a known problem whether a Sobolev homeomorphism between domains in R n can change sign of the Jacobian. The only case that remains open is whenand either f is Hölder continuous on almost all spheres of dimension [n/2], or f −1 is Hölder continuous on almost all spheres of dimensions n − [n/2] − 1, then the Jacobian of f is non-negative, J f ≥ 0, almost everywhere. This result is a consequence of a more general result proved in the paper. Here [x] stands for the greatest integer less than or equal to x.

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Section: Introduction and Resultsmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Lemma 18 (The Lebesgue Differentiation Theoremmentioning

confidence: 99%

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Jacobians of $W^{1,p}$ homeomorphisms, case $p=[n/2]$

Goldstein¹,

Hajłasz²

2018

Preprint

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“…Let us then concentrate ourselves on (4). Assume that x is a Lebesgue point for Du, and that J u (x) = 0; as a consequence, if we call y = u(x) then we have that Du −1 (y) exists and coincides with the inverse of Du(x).…”

Section: Preliminaries and Known Factsmentioning

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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“…Let us also recall that the Jacobian of a W 1,p , 1 ≤ p < n, Sobolev homeomorphism may behave strangely as it may vanish a.e. (see [22], [11] and [17]). As mentioned before the Jacobian of a homeomorphism cannot change sign if p > [n/2] by [24] and therefore the method of sign-changing Jacobian for providing a counterexample in Theorem 1.1 cannot be improved to p > [n/2].…”

Section: Introductionmentioning

confidence: 95%