  We develop new solvability methods for divergence form second order, real and
complex, elliptic systems above Lipschitz graphs, with $L_2$ boundary data. The
coefficients $A$ may depend on all variables, but are assumed to be close to
coefficients $A_0$ that are independent of the coordinate transversal to the
boundary, in the Carleson sense $\|A-A_0\|_C$ defined by Dahlberg. We obtain a
number of {\em a priori} estimates and boundary behaviour results under
finiteness of $\|A-A_0\|_C$. Our methods yield full characterization of weak
solutions, whose gradients have $L_2$ estimates of a non-tangential maximal
function or of the square function, via an integral representation acting on
the conormal gradient, with a singular operator-valued kernel. Also, the
non-tangential maximal function of a weak solution is controlled in $L_2$ by
the square function of its gradient. This estimate is new for systems in such
generality, and even for real non-symmetric equations in dimension 3 or higher.
The existence of a proof {\em a priori} to well-posedness, is also a new fact.
As corollaries, we obtain well-posedness of the Dirichlet, Neumann and
Dirichlet regularity problems under smallness of $\|A-A_0\|_C$ and
well-posedness for $A_0$, improving earlier results for real symmetric
equations. Our methods build on an algebraic reduction to a first order system
first made for coefficients $A_0$ by the two authors and A. McIntosh in order
to use functional calculus related to the Kato conjecture solution, and the
main analytic tool for coefficients $A$ is an operational calculus to prove
weighted maximal regularity estimates.
