Corrector estimates for elliptic systems with random periodic coefficients

TL;DR

This paper develops two methods to estimate the gradient of correctors in elliptic systems with random, stationary coefficients, providing bounds independent of domain size and applicable to both smooth and discontinuous coefficients.

Contribution

It introduces two approaches—Green function representation and spectral gap methods—for obtaining domain-size-independent moment bounds on corrector gradients in random elliptic systems.

Findings

Green function approach requires Hölder continuity and Logarithmic Sobolev inequality.

Spectral gap method applies to discontinuous coefficients with spectral gap estimates.

Bounds are independent of the domain size L.

Abstract

We consider an elliptic system of equations on the torus $\left[ -\frac{L}{2}, \frac{L}{2} \right)^d$ with random coefficients $A$, that are assumed to be coercive and stationary. Using two different approaches we obtain moment bounds on the gradient of the corrector, independent of the domain size $L$. In the first approach we use Green function representation. For that we require $A$ to be locally H\"older continuous and distribution of $A$ to satisfy Logarithmic Sobolev inequality. The second method works for non-smooth (possibly discontinuous) coefficients, and it requires that statistics of $A$ satisfies Spectral Gap estimate.