## Analysis & PDE

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

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

#### Abstract

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

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

Dates
Accepted: 18 November 2011
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513731263

Digital Object Identifier
doi:10.2140/apde.2012.5.983

Mathematical Reviews number (MathSciNet)
MR3022848

Zentralblatt MATH identifier
1275.35093

#### Citation

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. https://projecteuclid.org/euclid.apde/1513731263

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