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 $W^{2,q}(\Omega )\cap W_0^{1,s}(\Omega )$, $s\in (1,\infty )$, $q\in (1,n)$, 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.