The L^p Dirichlet problem for elliptic systems on Lipschitz domains

We develop a new approach to the L^p Dirichlet problem via L ² estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain Ω in ℝⁿ. For n > 4 and 2 - ε < p < 2(n-1)/n-3 + ε, we establish the solvability of the Dirichlet problem with boundary data in L^p(∂Ω). In the case of the polyharmonic equation Δ^lu = 0 with l ≥ 2, the range of p is sharp if 4 ≤ n < 2l + 1.