Uniform Lipschitz Estimates in Bumpy Half-Spaces

TL;DR

This paper establishes uniform Lipschitz and Hölder estimates near oscillating boundaries for elliptic systems with arbitrary boundary oscillations, enabling potential theoretic methods in complex boundary problems.

Contribution

It proves boundary regularity estimates without assuming any specific structure on boundary oscillations, broadening applicability in elliptic boundary value problems.

Findings

Uniform Lipschitz estimates near oscillating boundaries

Extension of potential theoretic methods to irregular boundaries

Implications for Green and Poisson kernel analysis

Abstract

This paper is devoted to the proof of uniform H\"older and Lipschitz estimates close to oscillating boundaries, for divergence form elliptic systems with periodically oscillating 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 stationary 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 theoretic methods in problems concerned with oscillating boundaries, which is an area of active research.