Necessary and sufficient conditions for the solvability of the L ^p dirichlet problem on Lipschitz domains

We study the homogeneous elliptic systems of order 2ℓ with real constant coefficients on Lipschitz domains in ℝ,n ≥ 4. For any fixed p > 2, we show that a reverse Hölder condition with exponent p is necessary and sufficient for the solvability of the Dirichlet problem with boundary data in L^p . We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the L^p Dirichlet problem for n ≥ 4 and 2 - ε < p < 2(n-1)/n-3 +ε. The range of p is known to be sharp if ℓ ≥ 2 and 4 ≤ n ≤ 2 ℓ + 1. For the polyharmonic equation, the sharp range of p is also found in the case n = 6, 7 if ℓ=2, and n=2 ℓ+2 if ℓ ≥ 3.