  We settle the issue of well-posedness for the Dirichlet problem for a higher
order elliptic system ${\mathcal L}(x,D_x)$ with complex-valued, bounded,
measurable coefficients in a Lipschitz domain $\Omega$, with boundary data in
Besov spaces.
  The main hypothesis under which our principal result is established is in the
nature of best possible and requires that, at small scales, the mean
oscillations of the unit normal to $\partial\Omega$ and of the coefficients of
the differential operator ${\mathcal L}(x,D_x)$ are not too large.
