Improved Regularity in Bumpy Lipschitz Domains

TL;DR

This paper proves Lipschitz regularity for solutions of elliptic systems with oscillating coefficients over highly oscillating Lipschitz boundaries, advancing understanding of boundary regularity without requiring smoother boundary conditions.

Contribution

It provides a new regularity result for elliptic systems on Lipschitz domains, improving previous estimates and introducing a boundary layer corrector estimate in the Sobolev-Kato class.

Findings

Lipschitz regularity achieved down to microscopic scales

Boundary layer corrector estimate established in Sobolev-Kato class

Improvement over previous regularity results in bumpy domains

Abstract

This paper is devoted to the proof of Lipschitz regularity, down to the microscopic scale, for solutions of an elliptic system with highly oscillating coefficients, over a highly oscillating Lipschitz boundary. The originality of this result is that it does not assume more than Lipschitz regularity on the boundary. Our Theorem, which is a significant improvement of our previous work on Lipschitz estimates in bumpy domains, should be read as an improved regularity result for an elliptic system over a Lipschitz boundary. Our progress in this direction is made possible by an estimate for a boundary layer corrector. We believe that this estimate in the Sobolev-Kato class is of independent interest.