A COUNTEREXAMPLE RELATED TO THE REGULARITY OF THE p-STOKES PROBLEM

摘要

We construct a solenoidal vector field u belonging to W 2 , q Ω ∩ W 0 1 , s Ω , q ∈ 1 n , s ∈ 1 ∞ $$ {W}^{2,q}left(Omega right)cap {W}_0^{1,s}left(Omega right),qin left(1,nright),sin left(1,infty right) $$ , 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.