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Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I
Université Paris-Sud.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
2011 (English)In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 184, no 1, 47-115 p.Article in journal (Refereed) Published
##### Abstract [en]

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.

##### Place, publisher, year, edition, pages
Springer , 2011. Vol. 184, no 1, 47-115 p.
##### Keyword [en]
elliptic systems, maximal regularity, Dirichlet and Neumann problems, square function, non-tangential maximal function, Carleson measure, functional and operational calculus
Mathematics
##### Identifiers
ISI: 000288674100002OAI: oai:DiVA.org:liu-63350DiVA: diva2:378797
Available from: 2010-12-16 Created: 2010-12-16 Last updated: 2012-01-03Bibliographically approved

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Axelsson Rosén, Andreas
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Applied MathematicsThe Institute of Technology
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