Higher-order linearization and regularity in nonlinear homogenization

TL;DR

This paper establishes large-scale smoothness and regularity results for solutions of nonlinear elliptic equations with random coefficients, extending classical homogenization theory to nonlinear settings with quantitative error estimates.

Contribution

It introduces a novel iterative approach to improve regularity, homogenization commutation, and error control for nonlinear elliptic equations with randomness.

Findings

Proves large-scale $C^ abla$ regularity for nonlinear elliptic solutions.

Provides a quantitative estimate on linearization errors.

Develops a nonlinear analogue of Taylor series with optimal remainder control.

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 $\bar{L}$, (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.