2018/05/01 by Scott N. Armstrong, Armstrong, Scott, Sam Ferguson +4 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1805.00467
We consider nonlinear, uniformly elliptic equations with random, highly\noscillating coefficients satisfying a finite range of dependence. We prove that\nhomogenization and linearization commute in the sense that the linearized\nequation (linearized around an arbitrary solution) homogenizes to the\nlinearization of the homogenized equation (linearized around the corresponding\nsolution of the homogenized equation). We also obtain a quantitative estimate\non the rate of this homogenization. These results lead to a better\nunderstanding of differences of solutions to the nonlinear equation, which is\nof fundamental importance in quantitative homogenization. In particular, we\nobtain a large-scale C0,1 estimate for differences of solutions---with\noptimal stochastic integrability. Using this estimate, we prove a large-scale\nC1,1 estimate for solutions, also with optimal stochastic integrability.\nEach of these regularity estimates are new even in the periodic setting. As a\nsecond consequence of the large-scale regularity for differences, we improve\nthe smoothness of the homogenized Lagrangian by showing that it has the same\nregularity as the heterogeneous Lagrangian, up to C2,1.\n