Bounded Continuous weak quasiregular mappings that fail to be quasiregular

TL;DR

This paper constructs bounded continuous weakly quasiregular mappings in dimensions n≥3 that are not quasiregular, demonstrating limitations of Sobolev regularity and the Jacobian sign condition in ensuring quasiregularity.

Contribution

It provides explicit examples of weakly quasiregular mappings that fail to be quasiregular below the critical Sobolev exponent, highlighting the limits of current regularity conditions.

Findings

Constructed bounded continuous weakly quasiregular mappings that are not quasiregular.

Showed that the Jacobian sign condition is too weak below W^{1,n} for orientation preservation.

Established that a one-sided distributional degree condition ensures quasiregularity for certain p ranges.

Abstract

We show that, in dimensions $n\geq 3$, continuity and boundedness do not restore the Sobolev regularity conjecture of Iwaniec and Martin for weakly quasiregular mappings below the critical exponent. For every bounded domain $\Omega\subset\mathbb R^n$ and every $1\leq p<nK/(K+1)$, we construct a bounded continuous weakly $K$-quasiregular mapping $$ f\in W^{1,\,p}(\Omega;\,\mathbb R^n)\cap C(\Omega;\,\mathbb R^n) \cap L^\infty(\Omega;\mathbb R^n) $$ which fails to be quasiregular. We further construct weakly quasiregular mappings whose singular sets have Hausdorff dimension arbitrarily close to the maximal size permitted by their Sobolev regularity. These examples show that, the almost-everywhere sign condition on the Jacobian is too weak to serve as an orientation-preserving hypothesis below $W^{1,n}$. In contrast, we show that, for $n-1<p<n$, quasiregularity follows once this condition is replaced by a one-sided condition on the distributional degree (together with boundedness).