A sense‐preserving Sobolev homeomorphism with negative Jacobian almost everywhere

Abstract

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