Sobolev homeomorphism that cannot be approximated by diffeomorphisms in $W^{1,1}$

Abstrakt

We construct a Sobolev homeomorphism in dimension $n\geq 4$, $f\in W^{1,1}((0,1)^n,\er^n)$ such that $J_f=\det Df>0$ 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 $W^{1,1}_{\loc}$.

Klíčová slova

• Sobolev homeomorphism
• Approximation