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

J.L. Fischer, F. Otto, Communications in Partial Differential Equations 41 (2016) 1108–1148.


Journal Article | Published
Author
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.
Publishing Year
Date Published
2016-07-02
Journal Title
Communications in Partial Differential Equations
Volume
41
Issue
7
Page
1108 - 1148
IST-REx-ID

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:10.1080/03605302.2016.1179318
Fischer, J. L., & Otto, F. (2016). A higher-order large scale regularity theory for random elliptic operators. Communications in Partial Differential Equations, 41(7), 1108–1148. https://doi.org/10.1080/03605302.2016.1179318
Fischer, Julian L, and Felix Otto. “A Higher-Order Large Scale Regularity Theory for Random Elliptic Operators.” Communications in Partial Differential Equations 41, no. 7 (2016): 1108–48. 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, pp. 1108–1148, 2016.
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.
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.

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