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

TL;DR

The paper proves the non-existence of certain Sobolev and Hölder homeomorphisms with low-rank gradients under specific smoothness conditions, highlighting a distinction between pointwise and distributional Jacobians.

Contribution

It establishes new non-existence results for Sobolev and Hölder homeomorphisms with low-rank gradients in the distributional sense, extending previous almost-everywhere results.

Findings

No Sobolev homeomorphisms with low-rank gradients exist under specified smoothness conditions.

No Hölder homeomorphisms with low-rank gradients exist under the same conditions.

Results clarify the limitations of low-rank Jacobians in Sobolev and Hölder homeomorphisms.

Abstract

We show that for any $k$ and $s > \frac{k+1}{k+2}$ there exist neither $W^{s,\frac{k}{s}}$-Sobolev nor $C^s$-H\"older 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.