A counterexample related to the regularity of the $p$-Stokes problem

TL;DR

This paper constructs a specific solenoidal vector field demonstrating that certain weighted regularity assumptions for the $p$-Stokes problem do not hold, challenging previous regularity results based on Muckenhoupt weights.

Contribution

It provides a counterexample showing the failure of the Korn inequality in weighted spaces for the $p$-Stokes problem, highlighting limitations in current regularity theory.

Findings

Constructed a solenoidal vector field in specified Sobolev spaces.

Showed that a particular weight does not belong to the Muckenhoupt class $A_ Infty$.

Demonstrated the failure of the Korn inequality in this context.

Abstract

In this paper we construct a solenoidal vector field $\mathbf{u}$ belonging to $W^{2,q}(\Omega)\cap W^{1,s}_0(\Omega)$, $s \in (1,\infty)$, $q \in (1,n)$, such that $(1+|\mathbf{Du}|)^{p-2}$, $p\in (1,2)\cup(2,\infty)$, does not belong to the Muckenhoupt class $A_\infty(\Omega)$. Thus, one cannot use the Korn inequality in weighted Lebesgue spaces to prove the natural regularity of the $p$-Stokes problem.