Uniform Lipschitz Estimates in Bumpy Half-Spaces

Abstract

This paper is devoted to the proof of uniform Hölder 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.

54 pages

References in corpus (8)

• Convergence Rates in L^2 for Elliptic Homogenization Problems
• The Green function estimates for strongly elliptic systems of second order
• Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains
• The Navier wall law at a boundary with random roughness
• Well-posedness of the Stokes-Coriolis system in the half-space over a rough surface
• Water waves over a rough bottom in the shallow water regime
• Homogenization of Green and Neumann Functions
• Uniform Regularity Estimates in Parabolic Homogenization

Cited by in corpus (10)

• Uniform boundary regularity in almost-periodic homogenization
• Uniform Regularity Estimates in Homogenization Theory of Elliptic System with Lower Order terms
• Higher-order boundary layers and regularity for Stokes systems over rough boundaries
• Uniform Integrability in Periodic Homogenization of Fully Nonlinear Equations
• Uniform Estimates in Periodic Homogenization of Fully Nonlinear Elliptic Equations
• Large-scale regularity for the stationary Navier-Stokes equations over non-Lipschitz boundaries
• Uniform Boundary Estimates in Homogenization of Higher Order Elliptic Systems
• Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law
• Regularity theory of elliptic systems in $\epsilon$-scale flat domains
• $L^2$-estimates for singular oscillatory integral operators