Vol. 5, No. 5, 2012

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Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems, II

Pascal Auscher and Andreas Rosén

Vol. 5 (2012), No. 5, 983–1061

We continue the development, by reduction to a first-order system for the conormal gradient, of L2 a priori estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence-form second-order complex elliptic systems. We work here on the unit ball and more generally its bi-Lipschitz images, assuming a Carleson condition as introduced by Dahlberg which measures the discrepancy of the coefficients to their boundary trace near the boundary. We sharpen our estimates by proving a general result concerning a priori almost everywhere nontangential convergence at the boundary. Also, compactness of the boundary yields more solvability results using Fredholm theory. Comparison between classes of solutions and uniqueness issues are discussed. As a consequence, we are able to solve a long standing regularity problem for real equations, which may not be true on the upper half-space, justifying a posteriori a separate work on bounded domains.

elliptic system, conjugate function, maximal regularity, Dirichlet and Neumann problems, square function, nontangential maximal function, functional and operational calculus, Fredholm theory
Mathematical Subject Classification 2010
Primary: 42B25, 35J56, 35J57, 35J25, 35J55
Received: 23 January 2011
Accepted: 18 November 2011
Published: 29 December 2012
Pascal Auscher
Département de Mathématiques d’Orsay
Université Paris-Sud et UMR 8628 du CNRS
Bâtiment 425
91405 Orsay Cedex
Andreas Rosén
Matematiska institutionen
Linköpings universitet
SE-581 83 Linköping