  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.
