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A higher-order large-scale regularity theory for random elliptic operators

Banff International Research Station for Mathematical Innovation and Discovery

We develop a large-scale regularity theory of higher or- der for divergence-form elliptic equations with heterogeneous coefficient fields a in the context of stochastic homogenization. Under the assump- tions of stationarity and slightly quantified ergodicity of the ensemble, we derive a Ck,α-“excess decay” estimate on large scales and a Ck,α- Liouville principle for any k ≥ 2: For a given a-harmonic function u on a ball BR, we show that its energy distance on some ball Br to the space of a-harmonic functions that grow at most like a polynomial of degree k has the natural decay in the radius r, at least above some minimal (random) radius r0. Our Liouville principle states that the space of a-harmonic functions that grow at most like a polynomial of degree k has (almost surely) the same dimension as in the constant-coefficient case. Our results rely on the existence of higher-order correctors for the homogenization problem, which we establish by an iterative construction.

Mathematics; Partial differential equations; Calculus of variations and optimal control; optimization

video/mp4

Author affiliation: Max Planck Institute for Mathematics in the Sciences

2016-03-11

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/57145

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/