TY - JOUR
T1 - Homogenization, Linearization, and Large-Scale Regularity for Nonlinear Elliptic Equations
AU - Armstrong, Scott
AU - Ferguson, Samuel J.
AU - Kuusi, Tuomo
N1 - Publisher Copyright:
© 2020 Wiley Periodicals LLC.
PY - 2021/2
Y1 - 2021/2
N2 - 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 C0, 1 estimate for differences of solutions—with optimal stochastic integrability. Using this estimate, we prove a large-scale C1, 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 C2, 1.
AB - 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 C0, 1 estimate for differences of solutions—with optimal stochastic integrability. Using this estimate, we prove a large-scale C1, 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 C2, 1.
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U2 - 10.1002/cpa.21902
DO - 10.1002/cpa.21902
M3 - Article
AN - SCOPUS:85084827088
SN - 0010-3640
VL - 74
SP - 286
EP - 365
JO - Communications on Pure and Applied Mathematics
JF - Communications on Pure and Applied Mathematics
IS - 2
ER -