Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I
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We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with L2 boundary data. The coefficients A may depend on all variables, but are assumed to be close to coefficients A0 that are independent of the coordinate transversal to the boundary, in the Carleson sense ‖A−A0‖C defined by Dahlberg. We obtain a number of a priori estimates and boundary behaviour results under finiteness of ‖A−A0‖C. Our methods yield full characterization of weak solutions, whose gradients have L2 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 L2 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−A0‖C and well-posedness for A0, improving earlier results for real symmetric equations. Our methods build on an algebraic reduction to a first order system first made for coefficients A0 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.
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