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 Versions disponibles : v1 (24-11-2009) v2 (25-08-2010) v3 (31-08-2010)
 Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I
 (24/11/2009)
 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 {\em 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 {\em 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.
 1 : Laboratoire de Mathématiques d'Orsay (LM-Orsay) CNRS : UMR8628 – Université Paris XI - Paris Sud 2 : Matematiska institutionen Stockholms universitet
 Domaine : Mathématiques/Equations aux dérivées partiellesMathématiques/Analyse fonctionnelle
 Mots Clés : Elliptic systems – Maximal regularity – Dirichlet and Neumann problems – Square function – Non-tangential maximal function – Carleson measure – Functional and operational calculus
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 hal-00435467, version 2 http://hal.archives-ouvertes.fr/hal-00435467 oai:hal.archives-ouvertes.fr:hal-00435467 Contributeur : Pascal Auscher <> Soumis le : Mercredi 25 Août 2010, 15:42:48 Dernière modification le : Mercredi 25 Août 2010, 16:27:17