Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II

TL;DR

This paper advances the understanding of boundary value problems for complex elliptic systems with non-smooth coefficients by developing new estimates and solvability results on Lipschitz domains, including a solution to a longstanding regularity problem.

Contribution

It introduces refined weighted maximal regularity estimates and solvability results for elliptic systems with non-smooth coefficients on Lipschitz domains, extending previous work to more general settings.

Findings

Established $L^2$ a priori estimates for elliptic systems.

Proved boundary non-tangential convergence results.

Solved a longstanding regularity problem for real elliptic equations.

Abstract

We continue the development, by reduction to a first order system for the conormal gradient, of $L^2$ \textit{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 \textit{a priori} almost everywhere non-tangential 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 \textit{a posteriori} a separate work on bounded domains.