A Counterexample Related to the Regularity of the p-Stokes Problem

Abstract

We construct a solenoidal vector field u belonging to \( {W}^{2,q}\left(\Omega \right)\cap {W}_0^{1,s}\left(\Omega \right),q\in \left(1,n\right),s\in \left(1,\infty \right) \), such that (1 + |Du|)^p − 2, p ∈ (1, ∞), p ≠ 2, does not belong to the Muckenhoupt class A_∞(Ω). Thus, one cannot use the Korn inequality in weighted Lebesgue spaces to prove the natural regularity of the p-Stokes problem.