We consider nonlinear, uniformly elliptic equations with random, highly oscillating coefficients satisfying a finite range of dependence. We prove that homogenization and linearization commute in the sense that the linearized equation (linearized around an arbitrary solution) homogenizes to the linearization of the homogenized equation (linearized around the corresponding solution of the homogenized equation). We also obtain a quantitative estimate on the rate of this homogenization. These results lead to a better understanding of differences of solutions to the nonlinear equation, which is of fundamental importance in quantitative homogenization. In particular, we obtain a large-scale $C^{0,1}$ estimate for differences of solutions---with optimal stochastic integrability. Using this estimate, we prove a large-scale $C^{1,1}$ estimate for solutions, also with optimal stochastic integrability. Each of these regularity estimates are new even in the periodic setting. As a second consequence of the large-scale regularity for differences, we improve the smoothness of the homogenized Lagrangian by showing that it has the same regularity as the heterogeneous Lagrangian, up to $C^{2,1}$.

中文翻译：

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非线性椭圆方程的均质化、线性化和大规模正则化

我们考虑非线性、均匀椭圆方程，其具有满足有限依赖范围的随机、高振荡系数。我们证明了均质化和线性化在线性化方程（围绕任意解线性化）均质化为均质化方程的线性化（围绕均质化方程的相应解线性化）的意义上交换。我们还获得了对这种同质化率的定量估计。这些结果有助于更好地理解非线性方程的解的差异，这在定量均质化中具有根本重要性。特别是，我们获得了对解决方案差异的大规模 $C^{0,1}$ 估计——具有最佳随机可积性。使用这个估计，我们证明了一个大规模的 $C^{1, 1}$ 对解决方案的估计，也具有最佳的随机可积性。即使在周期性设置中，这些规律性估计中的每一个都是新的。作为差异的大规模正则性的第二个结果，我们通过表明它与异质拉格朗日具有相同的正则性来提高均质拉格朗日的平滑度，最高可达 $C^{2,1}$。