Homogenization, linearization and large-scale regularity for nonlinear elliptic equations

2018/05/01 by Scott Armstrong, Scott N. Armstrong, Sam Ferguson +5 · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1805.00467

68 pages. We have added a new result, Theorem 1.3, giving a large-scale C^{1,1} estimate. We have also relaxed the regularity assumptions and obtained sharper regularity for the homogenized Lagrangian. Some references have been added and the introduction has been modified

arxiv created 2019/09/24 · arxiv updated 2019/09/26

Abstract

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.

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