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

Julian Fischer,Felix Otto

doi:10.1080/03605302.2016.1179318

Abstract

ABSTRACTWe develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fields a in the context of stochastic homogenization. The large-scale regularity of a-harmonic functions is encoded by Liouville principles: The space of a-harmonic functions that grow at most like a polynomial of degree k has the same dimension as in the constant-coefficient case. This result can be seen as the qualitative side of a large-scale Ck,α-regularity theory, which in the present work is developed in the form of a corresponding Ck,α-“excess decay” estimate: For a given a-harmonic function u on a ball BR, its energy distance on some ball Br to the above space of a-harmonic functions that grow at most like a polynomial of degree k has the natural decay in the radius r above some minimal radius r0.Though motivated by stochastic homogenization, the contribution of this paper is of purely deterministic nature: We work under the assumption that for the given realization a of the coefficient field, the couple (φ, σ) of scalar and vector potentials of the harmonic coordinates, where φ is the usual corrector, grows sublinearly in a mildly quantified way. We then construct “kth-order correctors” and thereby the space of a-harmonic functions that grow at most like a polynomial of degree k, establish the above excess decay, and then the corresponding Liouville principle.

Highlights

We are interested in the regularity of harmonic functions u associated with a uniformly elliptic coefficient field a in d space dimensions via the divergence-form equation
The proof of our large-scale Ck,α regularity theory relies in an essential way on the existence of kth-order correctors for the homogenization problem, which enable us to correct ahom-harmonic polynomials of degree k by adding a small perturbation
), into an a-harmonic function u with the same growth behavior is motivated by homogenization: We consider P as the “homogenized solution of the problem solved by u”, so that we think in terms of the two-scale expansion u ≈ P + φk∂kP and have that the error ψP := u − (P + φk∂kP ) satisfies

Summary

Introduction

The proof of our large-scale Ck,α regularity theory relies in an essential way on the existence of kth-order correctors for the homogenization problem, which enable us to correct ahom-harmonic polynomials of degree k by adding a small (in the L2-sense) perturbation. Under the assumption that we already have constructed an appropriate kth-order corrector on a ball BR, we show a Ck,α excess-decay estimate on large scales within this ball for a-harmonic functions (Lemma 14).

Results

Conclusion