Modulus of continuity of orientation preserving approximately differentiable homeomorphisms with a.e. negative Jacobian

Abstract

We construct an a.e. approximately differentiable homeomorphism of a unit n-dimensional cube onto itself which is orientation preserving, has the Lusin property (N) and has the Jacobian determinant negative a.e. Moreover, the homeomorphism together with its inverse satisfy a rather general sub-Lipschitz condition, in particular it can be bi-Hölder continuous with an arbitrary exponent less than 1.2010 Mathematics Subject Classification. Primary 46E35; Secondary 26B05, 26B10, 26B35, 74B20.

A Measure and Orientation Preserving Homeomorphism with Approximate Jacobian Equal −1 Almost Everywhere

“…It would be much easier to prove the result without conditions (b), (c), and with the condition (d) replaced by ap DΦ < 0 a.e. (see also [13]). However, in order to prescribe the derivative as in (1.2) we had to use deep results of Dacorogna and Moser [8] on the existence of diffeomorphisms with the prescribed Jacobian.…”

A sense‐preserving Sobolev homeomorphism with negative Jacobian almost everywhere

“…Thus, one is, naturally, led to Sobolev homeomorphisms and to questions about their Jacobian. This is the essence of the questions promoted by Hajlasz; see, for example, Goldstein and Hajlasz [10, 11]. Question Let $$Q\subset \mathbb {R}^n$$ be the open unit cube.…”

“…Thus, one is, naturally, led to Sobolev homeomorphisms and to questions about their Jacobian. This is the essence of the questions promoted by Hajlasz; see, for example, Goldstein and Hajlasz [10,11]. (a) Does there exist a homeomorphism 𝑓 ∈ 𝑊 1,𝑝 (𝑄, ℝ 𝑛 ) with 𝐽 𝑓 = det 𝐷𝑓 positive on a set of positive measure in 𝑄 and negative on a set of positive measure in 𝑄?…”

A sense preserving Sobolev homeomorphism with negative Jacobian almost everywhere

“…In 2001 Haj lasz posed a series of questions about the Jacobians of homeomorphisms which have some kind of derivative (weak or approximative). These questions appeared in several lecture notes and were recently reprinted in [11]. The essence of the questions can be summarized as follows:…”

“…The combination of Theorem 1.2 and Theorem 1.3 answers, up to the critical case p = [n/2], Question 1.1 a). Let us note that constructions of almost everywhere approximately differentiable homeomorphisms with everywhere negative (approximate) Jacobian are to be found in Goldstein and Haj lasz [9] and [11]. These maps lack the Sobolev regularity but have other striking properties e.g.…”