Necessary and sufficient conditions for the solvability of the Lp Dirichlet problem on Lipschitz domains

We study the homogeneous elliptic systems of order 2l with real constant coefficients on Lipschitz domains in R-n, n >= 4. For any fixed p > 2, we show that a reverse Holder 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 - epsilon < p < 2(n-1)/n-3 + epsilon. The range of p is known to be sharp if l >= 2 and 4 <= n = 2l + 1. For the polyharmonic equation, the sharp range of p is also found in the case n = 6, 7 if l =2, and n =2l + 2 if l >= 3.