Abstract
We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on mathbb {R}^d with stationary law (that is spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale varepsilon >0, we establish homogenization error estimates of the order varepsilon in case dgeqq 3, and of the order varepsilon |log varepsilon |^{1/2} in case d=2. Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence varepsilon ^delta . We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order (L/varepsilon )^{-d/2} for a representative volume of size L. Our results also hold in the case of systems for which a (small-scale) C^{1,alpha } regularity theory is available.
Highlights
In the present work, we establish quantitative homogenization results with optimal rates for nonlinear elliptic PDEs of the form−∇ · ( Aε(x, ∇uε) = f in Rd, (1)where Aε is a random monotone operator whose correlations decay quickly on scales larger than a microscopic scale ε
Our results may be seen as the optimal quantitative counterpart in the case of 2-growth to the qualitative stochastic homogenization theory for monotone systems developed by Dal Maso and Modica [17,18], and as the nonlinear counterpart of the optimal-order stochastic homogenization theory for linear elliptic equations developed by Gloria and Otto [32,33] and Gloria, Otto, and the second author [30,31]
Before stating our main results and the precise setting, let us introduce the key objects in the homogenization of nonlinear elliptic PDEs and systems with monotone nonlinearity
Summary
We establish quantitative homogenization results with optimal rates for nonlinear elliptic PDEs of the form. The first – logarithmic – rates of convergence in the stochastic homogenization of a nonlinear second-order elliptic PDE were obtained by Caffarelli and Souganidis [15] in the setting of non-divergence form equations. For forced mean curvature flow, Armstrong and Cardaliaguet [1] have derived a convergence rate of order ε1/90 These rates of convergence are all expected to be non-optimal (compare, for instance, the result for Hamilton-Jacobi equations to the rate of convergence ε known in the periodic homogenization setting [37]). In periodic homogenization of nonlinear elasticity the single-cell formula is valid for small deformations [40,41], and rates of convergence may be derived
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