The LpLp regularity problem for the Stokes system on Lipschitz domains

We study the LpLp regularity problem for the stationary Stokes system on Lipschitz domains. For any p>2p>2 we show that a weak reverse Hölder inequality with exponent p   is both necessary and sufficient for the solvability of the regularity problem with data in W1,p(∂Ω)∩Lnp(∂Ω). We also obtain the W1,p(Ω)W1,p(Ω) estimate ‖∇u‖Lp(Ω)+‖q‖Lp(Ω)≤‖f‖Lp(Ω)‖∇u‖Lp(Ω)+‖q‖Lp(Ω)≤‖f‖Lp(Ω) in a bounded Lipschitz domain Ω⊂RdΩ⊂Rd for solutions to the Dirichlet problem Δu=∇q+div(f)Δu=∇q+div(f), div(u)=0div(u)=0 in Ω and u=0u=0 on ∂Ω, where |1p−12|<12d+ε.