Approximation of W1, Sobolev homeomorphism by diffeomorphisms and the signs of the Jacobian

Abstract

Let Ω ⊂ R n , n ≥ 4, be a domain and 1 ≤ p < [n/2], where [a] stands for the integer part of a. We construct a homeomorphism f ∈ W 1,p ((−1, 1) n , R n ) such that J f = det Df > 0 on a set of positive measure and J f < 0 on a set of positive measure. It follows that there are no diffeomorphisms (or piecewise affine homeomorphisms) f k such that f k → f in W 1,p .

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

“…Here and in what follows, by an increasing function we mean a strictly increasing function. The Lipschitz modulus of continuity φ(t) = Ct does not satisfy the condition (2). However, φ(t) = Ct α , α ∈ (0, 1), satisfies both conditions.…”

“…The conditions (1), (2) and (3) are the same as in Theorem 1.2 and conditions (a), (b) are the same as conditions (1), (2) in Proposition 1.3. However, (c) means that ψ(t) is much smaller than φ(t) when t > 0 is sufficiently small.…”

“…However, φ(t) = Ct α , α ∈ (0, 1), satisfies both conditions. Also, there is a function φ that equals φ(t) = t log 2 t near zero and has both properties (1) and (2). A function with this modulus of continuity is Hölder continuous with any exponent α < 1.…”

“…Also, for the integration by parts to be rigorous, we should integrate from ε to 1 and then let ε → 0 + . Now we will present some technical lemmata related to functions φ satisfying (1) and (2). They will be needed later in the proof of Proposition 1.3.…”

Jacobian of weak limits of Sobolev homeomorphisms

“…Recently it was shown in [12] and [5] that a Jacobian of a Sobolev homeomorphism can change sign in dimension n ≥ for ≤ p < [ n ].…”