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. 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.