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 ⁡ D f > 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 .