Analysis & PDE

  • Anal. PDE
  • Volume 5, Number 5 (2012), 983-1061.

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

Pascal Auscher and Andreas Rosén

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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.

Article information

Anal. PDE, Volume 5, Number 5 (2012), 983-1061.

Received: 23 January 2011
Accepted: 18 November 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B25: Maximal functions, Littlewood-Paley theory 35J56: Boundary value problems for first-order elliptic systems 35J57: Boundary value problems for second-order elliptic systems 35J25: Boundary value problems for second-order elliptic equations 35J55

elliptic system conjugate function maximal regularity Dirichlet and Neumann problems square function nontangential maximal function functional and operational calculus Fredholm theory


Auscher, Pascal; Rosén, Andreas. Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems, II. Anal. PDE 5 (2012), no. 5, 983--1061. doi:10.2140/apde.2012.5.983.

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