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

Fischer JL, Otto F. 2016. A higher-order large scale regularity theory for random elliptic operators. Communications in Partial Differential Equations. 41(7), 1108–1148.

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Fischer, Julian L

^{ISTA}^{}; Otto, FelixAbstract

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.

Publishing Year

Date Published

2016-07-02

Journal Title

Communications in Partial Differential Equations

Volume

41

Issue

7

Page

1108 - 1148

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### Cite this

Fischer JL, Otto F. A higher-order large scale regularity theory for random elliptic operators.

*Communications in Partial Differential Equations*. 2016;41(7):1108-1148. doi:10.1080/03605302.2016.1179318Fischer, J. L., & Otto, F. (2016). A higher-order large scale regularity theory for random elliptic operators.

*Communications in Partial Differential Equations*. Taylor & Francis. https://doi.org/10.1080/03605302.2016.1179318Fischer, Julian L, and Felix Otto. “A Higher-Order Large Scale Regularity Theory for Random Elliptic Operators.”

*Communications in Partial Differential Equations*. Taylor & Francis, 2016. https://doi.org/10.1080/03605302.2016.1179318.J. L. Fischer and F. Otto, “A higher-order large scale regularity theory for random elliptic operators,”

*Communications in Partial Differential Equations*, vol. 41, no. 7. Taylor & Francis, pp. 1108–1148, 2016.Fischer, Julian L., and Felix Otto. “A Higher-Order Large Scale Regularity Theory for Random Elliptic Operators.”

*Communications in Partial Differential Equations*, vol. 41, no. 7, Taylor & Francis, 2016, pp. 1108–48, doi:10.1080/03605302.2016.1179318.**All files available under the following license(s):**

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