Exponential Decay of the Vorticity in the Steady-State Flow of a Viscous Liquid Past a Rotating Body

We construct a Sobolev homeomorphism in dimension \({n \geqq 4,\,f \in W^{1,1}((0, 1)^n,\mathbb{R}^n)}\) such that \({J_f = {\rm det} Df > 0}\) on a set of positive measure and J _f  < 0 on a set of positive measure. It follows that there are no diffeomorphisms (or piecewise affine homeomorphisms) f _k such that \({f_k\to f}\) in \({W^{1,1}_{\rm loc}}\).