Quantitative nonlinear homogenization: control of oscillations

Abstract

In this contribution we prove optimal rates of convergence in stochastic homogenization of monotone operators with p-growth for all 2 ≤ p < ∞ in dimensions d ≤ 3 (and in periodic homogenization for all p ≥ 2 and d ≥ 1). Previous contributions were so far restricted to p = 2. The main issues to treat superquadratic 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.

Quantitative hydrodynamic limits of the Langevin dynamics for gradient interface models

“…In this article, we are interested in studying this problem in the light of the recent progress in quantitative stochastic homogenization for nonlinear equations [7,6,41,27]. The interplay between homogenization and gradient interface models goes back to the work of [70], and quantitative homogenization methods was the main strategy used in [12].…”

“…• The parabolic equation (1.6) corresponds to the linearized equation in [7,6,41,27], that is, the Langevin dynamics linearized around the trajectories. The identification of the correlation structure and proof of the scaling limit for the Gibbs measure turns out to be equivalent to a homogenization statement for this linearized equation.…”

“…As stated, this strategy runs into a difficulty. First, as it has been recently observed in the setting of nonlinear elliptic homogenization in [7,6,41,27], in order to implement the two scale expansion (1.35), and obtain the optimal rate of convergence, one would need to have good control over the corrector associated with the linearized equation, defined, for a pair of slopes p, q ∈ R d , to be the map ψ p,q : (0, ∞) × Z d → R solution of the parabolic equation…”

“…The standard problem of stochastic homogenization of nonlinear elliptic equations, following [11,7,6,41,27], is defined as follows. We consider a Lagrangian L : (x, p) → L(x, p) with x, p ∈ R d and assume that for any x ∈ R d , the map p → L(x, p) is uniformly convex.…”

“…This latter quantity is difficult to control which is an obstruction to the extension of the theory. We refer to [27,Section 1.4] where the question is discussed in more details.…”