Quantitative Stochastic Homogenization and Large-Scale Regularity

Ferguson

Arch Rational Mech Anal

2020

Self Cite

We prove large-scale C ∞ 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 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 polynomiallygrowing 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. 65 6. Sharp estimates of linearization errors 71 7. Liouville theorems and higher regularity 72 Appendix A. Deterministic regularity estimates 80 Appendix B. Differentiation of F m 82 Appendix C. Linearization errors 83 Appendix D. Regularity for constant coefficient linearized equations 87 Appendix E. C ∞ regularity for smooth constant-coefficient Lagrangians 92 References 95

Section: 2supporting

confidence: 87%

Section: 2mentioning

confidence: 97%

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

“…With this definition, we lose the homogeneity of polynomials. Following the approach in [4,Chapter 3], it is natural to define heterogeneous versions of these spaces by…”

Section: 2mentioning

confidence: 99%

See 3 more Smart Citations

Higher-Order Linearization and Regularity in Nonlinear Homogenization

Ferguson

Arch Rational Mech Anal

2020

Self Cite

Homogenization, Linearization, and Large‐Scale Regularity for Nonlinear Elliptic Equations

Ferguson

2020

Comm Pure Appl Math

Self Cite

We consider nonlinear, uniformly elliptic equations with random, highly oscillating coefficients satisfying a finite range of dependence. We prove that homogenization and linearization commute in the sense that the linearized equation (linearized around an arbitrary solution) homogenizes to the linearization of the homogenized equation (linearized around the corresponding solution of the homogenized equation). We also obtain a quantitative estimate on the rate of this homogenization. These results lead to a better understanding of differences of solutions to the nonlinear equation, which is of fundamental importance in quantitative homogenization. In particular, we obtain a large‐scale C0, 1 estimate for differences of solutions—with optimal stochastic integrability. Using this estimate, we prove a large‐scale C1, 1 estimate for solutions, also with optimal stochastic integrability. Each of these regularity estimates are new even in the periodic setting. As a second consequence of the large‐scale regularity for differences, we improve the smoothness of the homogenized Lagrangian by showing that it has the same regularity as the heterogeneous Lagrangian, up to C2, 1. © 2020 Wiley Periodicals LLC

Large‐Scale Analyticity and Unique Continuation for Periodic Elliptic Equations

Smart

2020

Comm Pure Appl Math

We prove that a solution of an elliptic operator with periodic coefficients behaves on large scales like an analytic function in the sense of approximation by polynomials with periodic corrections. Equivalently, the constants in the largescale C k;1 estimate scale exponentially in k, just as for the classical estimate for harmonic functions, and the minimal scale grows at most linearly in k. As a consequence, we characterize entire solutions of periodic, uniformly elliptic equations that exhibit growth like O.exp.jxj// for small > 0. The large-scale analyticity also implies quantitative unique continuation results, namely a threeball theorem with an optimal error term as well as a proof of the nonexistence of L 2 eigenfunctions at the bottom of the spectrum.