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 aEuro-A-A (0)aEuro- (C) defined by Dahlberg. We obtain a number of a priori estimates and boundary behaviour results under finiteness of aEuro-A-A (0)aEuro- (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. As corollaries, we obtain well-posedness of the Dirichlet, Neumann and Dirichlet regularity problems under smallness of aEuro-A-A (0)aEuro- (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.