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 L 2 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.Andreas Rosén was formerly called Andreas Axelsson.

Locations

• Analysis & PDE - View - PDF
• arXiv (Cornell University) - View - PDF
• Project Euclid (Cornell University) - View - PDF
• HAL (Le Centre pour la Communication Scientifique Directe) - View - PDF