Armstrong Scott, Ferguson Samuel J, Kuusi Tuomo
1Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012 USA.
2Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, (Gustaf Hällströmin katu 2), 00014 Helsinki, Finland.
Arch Ration Mech Anal. 2020;237(2):631-741. doi: 10.1007/s00205-020-01519-1. Epub 2020 Apr 9.
We prove large-scale regularity for solutions of nonlinear elliptic equations with random coefficients, thereby obtaining a version of the statement of Hilbert's 19th problem in the context of homogenization. The analysis proceeds by iteratively improving three statements together: (i) the regularity of the homogenized Lagrangian , (ii) the commutation of higher-order linearization and homogenization, and (iii) large-scale -type regularity for higher-order linearization errors. We consequently obtain a quantitative estimate on the scaling of linearization errors, a Liouville-type theorem describing the polynomially-growing solutions of the system of higher-order linearized equations, and an explicit (heterogenous analogue of the) Taylor series for an arbitrary solution of the nonlinear equations-with the remainder term optimally controlled. These results give a complete generalization to the nonlinear setting of the large-scale regularity theory in homogenization for linear elliptic equations.
我们证明了具有随机系数的非线性椭圆方程解的大规模正则性，从而在均匀化背景下得到了希尔伯特第19问题陈述的一个版本。分析过程是通过一起迭代地改进三个陈述来进行的：（i）均匀化拉格朗日量的正则性，（ii）高阶线性化与均匀化的交换性，以及（iii）高阶线性化误差的大规模型正则性。因此，我们得到了关于线性化误差缩放的定量估计、一个描述高阶线性化方程组多项式增长解的刘维尔型定理，以及非线性方程任意解的显式（异质类似物）泰勒级数——其余项得到最优控制。这些结果对线性椭圆方程均匀化中的大规模正则性理论在非线性情形下进行了完全推广。
