The Neumann problem in Lp on Lipschitz and convex domains

Given p>2 and a bounded Lipschitz domain Ω, we obtain a necessary and sufficient condition for the solvability of the Neumann problem for Laplace's equation Δu=0 in Ω with boundary data in Lp(∂Ω). As a result, we show that the Lp Neumann problem on convex domains in Rd is solvable for 1