The Lp regularity problem for the Stokes system on Lipschitz domains

We study the L-p regularity problem for the stationary Stokes system on Lipschitz domains. For any p > 2 we show that a weak reverse Holder inequality with exponent p is both necessary and sufficient for the solvability of the regularity problem with data in W-1,W-P(partial derivative Omega) boolean AND L-n(p) (partial derivative Omega). We also obtain the W-1,W-P(Omega) estimate parallel to del u parallel to(LP) ((Omega)) + parallel to q parallel to(LP) ((Omega)) <= parallel to f parallel to(LP) ((Omega)) in a bounded Lipschitz domain Omega subset of R-d for solutions to the Dirichlet problem Delta u = del q + div(f), div(u) = 0 in Omega and u = 0 on partial derivative Omega, where vertical bar 1/p - 1/2 vertical bar < 1/2d + epsilon. (C) 2015 Elsevier Inc. All rights reserved.