A sense preserving Sobolev homeomorphism with negative Jacobian almost everywhere

Abstract

For every $1\le p<\frac{3}{2}$ we construct a Sobolev homeomorphism $f\in W^{1,p}([-1,1{]}^4,[-1,1{]}^4)$ such that $f(x)=x$ for every $x\in \partial [-1,1{]}^4$ but $J_f<0$ a.e.