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

By Julian Fischer, Felix Otto. 2015

51 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/26945450b9fb9eaa89f8a68cbe805f228f75f3e5

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In Applied Mathematics

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Citations in the corpus

Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization (2015, result): “on of the solution by the polynomial is, up to a constant, as good as one would have for a harmonic function with the same oscillation. This result is presented below in Theorem 2.1. Fischer and Otto [9] have very recently developed a higher regularity theory along somewhat diﬀerent lines (see also ”
The additive structure of elliptic homogenization (2016, background): “The papers [17, 14] were the first to write regularity estimates in the form of (3.”
Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, background): “Following [15], the regularity theory was expanded and completed rather rapidly in a series of papers by several authors [14, 63, 13, 52, 11, 12], where the higher regularity theory and related Liouville theorems were established.”