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

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.