  We study the homogeneous elliptic systems of order $2\ell$ with real constant
coefficients on Lipschitz domains in $R^n$, $n\ge 4$. For any fixed $p>2$, we
show that a reverse H\"older 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 $n\ge 4$ and
$2-\e< p<\frac{2(n-1)}{n-3} +\e$. The range of $p$ is known to be sharp if
$\ell\ge 2$ and $4\le n\le 2\ell +1$. For the polyharmonic equation, the sharp
range of $p$ is also found in the case $n=6$, 7 if $\ell=2$, and $n=2\ell+2$ if
$\ell\ge 3$.
