A Large-Scale Regularity Theory for Random Elliptic Operators on the Half-Space with Homogeneous Neumann Boundary Data

TL;DR

This paper establishes large-scale regularity results for solutions to random elliptic equations on the half-space with Neumann boundary conditions, extending previous work on Dirichlet conditions using stochastic homogenization techniques.

Contribution

It introduces a novel Neumann boundary-adapted corrector and extends the large-scale regularity theory to Neumann boundary data for elliptic operators with randomness.

Findings

Derived large-scale regularity properties for Neumann problems

Constructed a Neumann boundary-adapted corrector

Extended stochastic homogenization techniques to Neumann boundary conditions

Abstract

In this note we derive large-scale regularity properties of solutions to second-order linear elliptic equations with random coefficients on the half- space with homogeneous Neumann boundary data; it is a companion to arXiv:1604.02717 in which the situation for homogeneous Dirichlet boundary data was addressed. Similarly to arXiv:1604.02717, the results in this contribution are expressed in terms of a first-order Liouville principle. It follows from an excess-decay that is shown through means of a stochastic homogenization-inspired Campanato iteration. The core of this contribution is the construction of a sublinear half-space- adapted corrector/vector potential pair that, in contrast to arXiv:1604.02717, is adapted to the Neumann boundary data.