A Liouville theorem for elliptic systems with degenerate ergodic coefficients

TL;DR

This paper proves a Liouville theorem and regularity estimates for second-order degenerate elliptic systems with stationary ergodic random coefficients, advancing understanding of their large-scale behavior.

Contribution

It establishes an intrinsic large-scale regularity estimate and a Liouville theorem for degenerate elliptic systems with ergodic coefficients, under moment bounds.

Findings

Proved large-scale $C^{1,eta}$ regularity for $a$-harmonic functions.

Established a first-order Liouville theorem for subquadratic solutions.

Extended regularity theory to degenerate elliptic systems with random coefficients.

Abstract

We study the behavior of second-order degenerate elliptic systems in divergence form with random coefficients which are stationary and ergodic. Assuming moment bounds like Chiarini and Deuschel [Arxiv preprint 1410.4483, 2014] on the coefficient field $a$ and its inverse, we prove an intrinsic large-scale $C^{1,\alpha}$-regularity estimate for $a$-harmonic functions and obtain a first-order Liouville theorem for subquadratic $a$-harmonic functions.