An estimate for the solutions of stokes equations in exterior domains

Abstract

A boundary value problem for the Stokes equations is examined in an exterior domain Ω ⊂ℝnwith a uniform Dirichlet condition on the boundary and a homogeneous condition at infinity. It is shown that estimating the norm Lp(Ω) of the second derivatives of the velocity vector field by the same norm of the exterior forces vector field is correct for p < n/2, but not for p ≥ n/2. This estimate is valid also for p ≥ n/2 if the boundary conditions are modified at infinity.