Abstract
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.
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Auscher, P., Axelsson, A. Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I. Invent. math. 184, 47–115 (2011). https://doi.org/10.1007/s00222-010-0285-4
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DOI: https://doi.org/10.1007/s00222-010-0285-4