论文标题
退火的Calderon-Zygmund估计是定量随机均质化的便捷工具
The annealed Calderon-Zygmund estimate as convenient tool in quantitative stochastic homogenization
论文作者
论文摘要
本文介绍了以随机系数为单位形式的线性椭圆方程的定量均质化理论。我们得出了对均质化误差的梯度估计，即在强范围（振荡）和弱规范（波动）方面，实际解决方案和均质解决方案的两尺度扩展之间的差异。这些估计值在微观和宏观尺度之间的比率方面是最佳的。本文的目的是强调最近引入的退火Calderon-Zygmund（CZ）估计值，以获取上述以前已知的错误估计。此外，本文提供了这些退火CZ估计的新颖证明，这些估计完全避免了规律性理论，但依赖于功能分析。基于这样的观察，即使在操作员规范的水平上，Helmholtz投影都接近均质系数的投影（为此，很容易获得退火的CZ估计值）。在本文中，我们努力寻求简单的证据，从而将自己限制在系数的集合中，这些系数是具有可集成相关性和Hölder持续实现的高斯随机字段的局部变换。与较早的工作一样，我们使用了来自均质一般理论的天然对象，例如（势和通量）校正器和均质化换向器。振荡和波动估计均取决于敏感性计算，即估计利益量的敏感性如何取决于系数领域的无限变化，该系数被送入光谱间隙不等式。在本文中，退火的CZ估计值是椭圆规则理论进入的唯一形式。
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 Calderon-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.
