Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I

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 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 a priori to well-posedness, is also a new fact.