Abstract
We prove large-scale C^infty regularity for solutions of nonlinear elliptic equations with random coefficients, thereby obtaining a version of the statement of Hilbert’s 19th problem in the context of homogenization. The analysis proceeds by iteratively improving three statements together: (i) the regularity of the homogenized Lagrangian overline{L}, (ii) the commutation of higher-order linearization and homogenization, and (iii) large-scale C^{0,1}-type regularity for higher-order linearization errors. We consequently obtain a quantitative estimate on the scaling of linearization errors, a Liouville-type theorem describing the polynomially-growing solutions of the system of higher-order linearized equations, and an explicit (heterogenous analogue of the) Taylor series for an arbitrary solution of the nonlinear equations—with the remainder term optimally controlled. These results give a complete generalization to the nonlinear setting of the large-scale regularity theory in homogenization for linear elliptic equations.
Highlights
L is a stochastic object: it is sampled by a probability measure P which is statistically stationary and satisfies a unit range of dependence
The objective is to describe the statistical behavior of the solutions of (1.1), with respect to the probability measure P, on large length scales
Quantitative homogenization for the nonlinear equation (1.1) has a comparatively sparse literature; the only such results of which we are aware are those of [5,6], our previous paper [1] and a new paper of Fischer and Neukamm [14] which was posted to arXiv as we were finishing the present article
Summary
This article is concerned with nonlinear, divergence-form, uniformly elliptic equations of the form. We will prove large-scale Ck, type estimates for solutions of (1.1), for k ∈ N as large as can be expected from the regularity assumptions on L, a result analogous to Hilbert’s 19th problem, famously given for spatially homogeneous Lagrangians by De Giorgi and Nash. In the case of homogenization, the situation is reversed, as it is less clear how one should “differentiate the equation” since literally doing so would produce negative powers of ε, even if the coefficients were smooth Our analysis resolves this difficulty and reveals the interplay between these three seemingly different kinds of results: (i) the regularity of L, (ii) the homogenization of linearized equations and the commutability of homogenization and linearization, and (iii) the large-scale regularity of the solutions. These three statements must be proved together, iteratively in the parameter k ∈ N which represents the degree of regularity of L, the order of the linearized equation, and the order of the Ck, estimate
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