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


Author(s): Fischer, Julian; Otto, Felix
Article Title: A higher-order large scale regularity theory for random elliptic operators
Affiliation
Abstract: We 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.
Keywords: Ck,α regularity; higher-order correctors; Liouville principle; random elliptic operator; regularity theory; stochastic homogenization
Journal Title: Communications in Partial Differential Equations
Volume: 41
Issue 7
ISSN: 1532-4133
Publisher: Taylor & Francis  
Date Published: 2016-07-02
Start Page: 1108
End Page: 1148
URL:
DOI: 10.1080/03605302.2016.1179318
Open access: yes (repository)
IST Austria Authors
  1. Julian Fischer
    13 Fischer
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