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Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I

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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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Authors and Affiliations

  1. Univ. Paris-Sud, laboratoire de Mathématiques, UMR 8628, Orsay, 91405, France

    Pascal Auscher

  2. CNRS, Orsay, 91405, France

    Pascal Auscher

  3. Centre for Mathematics and Its Applications, Australian National University, Canberra, ACT, 0200, Australia

    Pascal Auscher

  4. Matematiska Institutionen, Linköpings Universitet, 581 83, Linköping, Sweden

    Andreas Axelsson

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  1. Pascal Auscher
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  2. Andreas Axelsson
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Correspondence to Pascal Auscher.

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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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