Higher-Order Linearization and Regularity in Nonlinear Homogenization
Higher-Order Linearization and Regularity in Nonlinear Homogenization
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DOI：
10.1007/s00205-020-01519-1
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发表时间：
2019-10
影响因子：
2.5
通讯作者：
S. Armstrong;Samuel J. Ferguson;Tuomo Kuusi
中科院分区：
文献类型：
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作者：
S. Armstrong;Samuel J. Ferguson;Tuomo Kuusi
我们证明了随机系数非线性椭圆型方程解的大尺度正则性，从而在均匀化的背景下得到了Hilbert第19问题的一个版本。分析通过迭代改进三个陈述来进行：（i）均匀化拉格朗日的正则性，（ii）高阶线性化和均匀化的交换，以及（iii）高阶线性化误差的大尺度型正则性。因此，我们得到一个定量估计的线性化误差的缩放，刘维尔型定理描述的多项式增长的解决方案的系统的高阶线性化方程，和一个明确的（异构模拟）泰勒级数的任意解的非线性方程的剩余项最优控制。这些结果完全推广到线性椭圆型方程均匀化大规模正则性理论的非线性情形。
We prove large-scaleregularity 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.