Quantitative nonlinear homogenization
Nicolas Clozeau, Antoine Gloria · arXiv (Cornell University) · 2021
In this contribution we prove optimal rates of convergence in stochastic homogenization of monotone operators with $p$-growth for all $2\le p <\infty$ in dimensions $d\le 3$ (and in periodic homogenization for all $p\ge 2$ and $d\ge 1$). Previous contributions were so far restricted to $p=2$. The main issues to treat super-quadratic growth are the potential degeneracy and the unboundedness of the coefficients when linearizing the equation. To deal with this we develop a perturbative regularity theory in the large for random linear operators with unbounded coefficients. From the probabilistic point of view we rely on functional inequalities, and make crucial use of the central limit theorem scaling to absorb nonlinear contributions. Combining these two ingredients we obtain sharp bounds on the growth of correctors and of corrector differences, from which the quantitative two-scale expansion follows. Our approach bypasses the need of non-perturbative large-scale regularity theory -- which might not hold true in the general setting we consider here.