## 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 {double pipe}A-A_{0}{double pipe}_{C} defined by Dahlberg. We obtain a number of a priori estimates and boundary behaviour results under finiteness of {double pipe}A-A_{0}{double pipe}_{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 {double pipe}A-A_{0}{double pipe}_{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.

Original language | English |
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Pages (from-to) | 47-115 |

Number of pages | 69 |

Journal | Inventiones Mathematicae |

Volume | 184 |

Issue number | 1 |

DOIs | |

Publication status | Published - Apr 2011 |