Jacobians of $W^{1,p}$ homeomorphisms, case $p=[n/2]$

Abstract

We investigate a known problem whether a Sobolev homeomorphism between domains in R n can change sign of the Jacobian. The only case that remains open is whenand either f is Hölder continuous on almost all spheres of dimension [n/2], or f −1 is Hölder continuous on almost all spheres of dimensions n − [n/2] − 1, then the Jacobian of f is non-negative, J f ≥ 0, almost everywhere. This result is a consequence of a more general result proved in the paper. Here [x] stands for the greatest integer less than or equal to x.

Linking topological spheres

“…Since the linking number of highly non-smooth topological spheres has recently been used in geometric analysis (cf. [6,8,7,12]), the author hopes that this example will be of some interest.…”