Homeomorphism with zero Jacobian: Sharp integrability of the derivative

Let N ⩾ 2 . We construct a homeomorphism f ∈ W 1 , 1 ( [ 0 , 1 ] N , R N ) such that J f = 0 almost everywhere and sup 0 < ε ⩽ N − 1 ε ∫ [ 0 , 1 ] N | D f | N − ε < ∞ . In particular, f ∈ W 1 , p ( [ 0 , 1 ] N , [ 0 , 1 ] N ) for all p ∈ [ 1 , N ) .