We continue the development, by reduction to a first-order system for the conormal gradient, of L2a 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.
This article investigates how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain in the case when the domains involved are of class C-1. We consider the Laplacian and use results developed previously for the corresponding Lipschitz case. In contrast with the Lipschitz case, however, in the C-1-case we derive an asymptotic formula for the eigenvalues when the domains are of class C-1. Moreover, as an application we consider the case of a C-1-perturbation when the reference domain is of class C-1,C-alpha.