The annealed Calderon-Zygmund estimate as convenient tool in quantitative stochastic homogenization
Marc Josien and Felix Otto
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Submission date: 19. May. 2020
MSC-Numbers: 35B27, 35R60, 35J15, 35B45, 42B15
Keywords and phrases: stochastic homogenization, Calderon-Zygmund estimates, corrector estimates, oscillations, fluctuations
Link to arXiv: See the arXiv entry of this preprint.
This article is about the quantitative homogenization theory of linear elliptic equations in divergence form with random coeﬃcients. We derive gradient estimates on the homogenization error, i.e. on the diﬀerence between the actual solution and the two-scale expansion of the homogenized solution, both in terms of strong norms (oscillation) and weak norms (ﬂuctuation). These estimates are optimal in terms of scaling in the ratio between the microscopic and the macroscopic scale.
The purpose of this article is to highlight the usage of the recently introduced annealed Calderón-Zygmund (CZ) estimates in obtaining the above, previously known, error estimates. Moreover, the article provides a novel proof of these annealed CZ estimate that completely avoids quenched regularity theory, but rather relies on functional analysis. It is based on the observation that even on the level of operator norms, the Helmholtz projection is close to the one for the homogenized coeﬃcient (for which annealed CZ estimates are easily obtained).
In this article, we strive for simple proofs, and thus restrict ourselves to ensembles of coeﬃcient ﬁelds that are local transformations of Gaussian random ﬁelds with integrable correlations and Hölder continuous realizations. As in earlier work, we use the natural objects from the general theory of homogenization, like the (potential and ﬂux) correctors and the homogenization commutator. Both oscillation and ﬂuctuation estimates rely on a sensitivity calculus, i.e. on estimating how sensitively the quantity of interest does depend on an inﬁnitesimal change in the coeﬃcient ﬁeld, which is fed into the Spectral Gap inequality. In this article, the annealed CZ estimate is the only form in which elliptic regularity theory enters.