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.

UR - http://www.scopus.com/inward/record.url?scp=85084827088&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85084827088&partnerID=8YFLogxK

U2 - 10.1002/cpa.21902

DO - 10.1002/cpa.21902

M3 - Article

AN - SCOPUS:85084827088

VL - 74

SP - 286

EP - 365

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 2

ER -