@article{1318,
  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.},
  author       = {Fischer, Julian L and Otto, Felix},
  issn         = {1532-4133},
  journal      = {Communications in Partial Differential Equations},
  keywords     = {Ck, α regularity, higher-ordercorrectors, Liouville principle, random elliptic operator, regularity theory, stochastic homogenization},
  number       = {7},
  pages        = {1108 -- 1148},
  publisher    = {Taylor & Francis},
  title        = {{A higher-order large scale regularity theory for random elliptic operators}},
  doi          = {10.1080/03605302.2016.1179318},
  volume       = {41},
  year         = {2016},
}