Approximation of W1,p 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 ∈ W1,p((−1, 1)n, R n) such that Jf = det Df > 0 on a set of positive measure and Jf < 0 on a set of positive measure. It follows that there are no diffeomorphisms (or piecewise affine homeomorphisms) fk such that fk → f in W1,p .