Uniform Lipschitz Estimates in Bumpy Half-Spaces

This paper is devoted to the proof of uniform Hölder and Lipschitz estimates close to oscillating boundaries, for divergence form elliptic systems with periodically oscil-lating coefficients. Our main point is that no structure is assumed on the oscillations of the boundary. In particular, those are neither periodic, nor quasiperiodic, nor sta-tionary ergodic. We investigate the consequences of our estimates on the large scales of Green and Poisson kernels. Our work opens the door to the use of potential theo-retic methods in problems concerned with oscillating boundaries, which is an area of active research.