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

Let Ω⊂Rn, 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,Rn) 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.