Construction of W^{2,n}(Ω) function with gradient violating Lusin (N) condition

Abstract

We construct the smooth function f, such that its derivative belongs to W^{1,n}((0,1)^n,R^n) and it maps a line onto the full unit square. This shows that a mapping which maps a set of zero measure onto the set of positive measure can be a gradient mapping.

Keywords

• Sobolev space
• Lusin condition
• Peano curve