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

Abstract

Abstract. We construct an almost everywhere approximately differentiable, orientation and measure preserving homeomorphism of a unit n-dimensional cube onto itself, whose Jacobian is equal to −1 a.e. Moreover we prove that our homeomorphism can be uniformly approximated by orientation and measure preserving diffeomorphisms.

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

“…One can show as in [7] that the sequence {F k } converges in the uniform metric (1.1) to a homeomorphism F that has all properties listed in Theorem 1.2, but the Lusin property (N). However, instead of referring to [7] we will use a straightforward argument showing convergence of a subsequence of {F k }. We proved in (4.9) that both families {F k } and {F −1 k } are equicontinuous.…”

“…The construction guarantees that |Q \ ∞ k=1 K k | = 0. The sequence of of homeomorphisms {F k } is a Cauchy sequence with respect to the uniform metric d. It is well known and easy to check that the space of homeomorphisms of Q onto itself is complete with respect to the uniform metric d (see [7,Lemma 1.2]) so the sequence {F k } converges to a homeomorphism F . Since |Q \ ∞ k=1 K k | = 0 and F k has negative Jacobian on K k , it follows that F has negative Jacobian almost everywhere.…”

“…The next lemma is similar to [7, Lemma 3.8], but some estimates are new. Let us recall the notation used in [7].…”

“…Let G : Ω → R n , Ω ⊂ R n , be a C ∞ -diffeomorphism such that Sketch of the proof. Arguments that are similar to those that appear in the proof of Lemma 3.8 in [7] will be sketched only; for details we refer the reader to [7].…”

“…(a) Φ| ∂Q = id , (b) Φ is measure preserving, (c) Φ is a limit, in the uniform metric d, of a sequence of measure preserving C ∞diffeomorphisms of Q that are identity on the boundary, (d) the approximate derivative of Φ satisfies Note that (b) implies the Lusin condition (N). The proof given in [7] does not give any estimates for the modulus of continuity of the homeomorphism Φ. One of our main concerns in Theorem 1.1 were the conditions (b) and (c).…”