Obstacles for Sobolev-homeomorphisms with low rank -- pointwise a.e. vs distributional Jacobians

Abstract

We show that for any $k$ and $s>\frac{k+1}{k+2}$ there exist neither $W^{s,\frac{k}{s}}$-Sobolev nor $C^s$-Hölder homeomorphisms from the disk ${\mathbb{B}}^n$ into ${\mathbb{R}}^N$ whose gradient has rank $<k$ in distributional sense. This complements known examples of such kind of homeomorphisms whose gradient has rank $<k$ almost everywhere.