The Lp Neumann problem for Laplace's equation on convex domains

On hounded Lipschitz domains in Rn, for any fixed p > 2, we show that a reverse Holder inequality with exponent p is necessary and sufficient for the solvability of the Neumann problem with boundary data in Lp. As a consequence, on bounded convex domains, we establish the solvability of the Lp Neumann problem for 1 < p < infinity, if n = 2, for 1 < p < 4, if n = 3 and for 1 < p < 3 + epsilon, if n ≥ 4.