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 HAL : hal-00435467, version 2
 Weighted maximal regularity estimates and solvability of non-smooth elliptic systems IAuscher P., Axelsson A.http://hal.archives-ouvertes.fr/hal-00435467
 Versions disponibles : v1 (24-11-2009) v2 (25-08-2010) v3 (31-08-2010)
Preprint, Working Paper, Document sans référence, etc.
 Mathématiques/Equations aux dérivées partielles Mathématiques/Analyse fonctionnelle
Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I
Pascal Auscher () 1, Andreas Axelsson () 2
 1 : Laboratoire de Mathématiques d'Orsay (LM-Orsay) http://www.math.u-psud.fr CNRS : UMR8628 – Université Paris XI - Paris Sud France 2 : Matematiska institutionen Stockholms universitet Suède
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.
Anglais

Elliptic systems – Maximal regularity – Dirichlet and Neumann problems – Square function – Non-tangential maximal function – Carleson measure – Functional and operational calculus
MSC classes: 35J55, 35J25, 42B25, 47N20
version largement révisée et augmentée de 10 pages suivant les remarques des rapporteurs. Une section présente un résumé de l'algorithme et un résultat de rigidité pour une certaine classe de solutions est énoncé.

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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