A higher-order large-scale regularity theory for random elliptic operators. Issue 7 (2nd July 2016) - Fischer, Julian; Otto, Felix
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 C k, α -regularity theory, which in the present work is developed in the form of a corresponding C k, α -"excess decay" estimate: For a given a -harmonic function u on a ball B R, its energy distance on some ball B r 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 r 0 . 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 " k th-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.