Construction of W2,n(Ω) function with gradient violating Lusin (N) condition

Abstract

We construct f∈C1([−1,1]n), such that Df∈W1,n((−1,1)n,Rn) maps [−1,1]×{0}n−1 onto [−1,1]n. This shows that a mapping which maps a set of zero measure onto the set of positive measure can be a gradient mapping.