Homeomorphism with zero Jacobian: Sharp integrability of the derivative

Let N⩾2. We construct a homeomorphism f∈W1,1(N[0,1],RN) such that Jf=0 almost everywhere and sup0<ε⩽N−1ε∫N[0,1]|Df|N−ε<∞. In particular, f∈W1,p(N[0,1],N[0,1]) for all p∈[1,N).