Graduate course M285K, University of California, Los Angeles, Winter 2021
Abstract: Homogenization theory is concerned with the large-scale statistical behavior of solutions to PDEs with heterogeneous coefficients. Its development was originally driven to provide a rigorous approach to in silico predictions for composite materials: such materials indeed give rise to PDEs with coefficients that vary on a length scale much smaller than the observation scale, and those oscillations make numerical methods too costly. In many cases, the solution operator is expected to behave on large scales like the one for an "effective" problem with homogeneous coefficients, leading to a huge reduction in complexity. Homogenization theory aims to justify this type of useful averaging result and to provide representation formulas (or approximations) for effective coefficients. The theory was further developed to quantify the convergence rate for solutions of heterogeneous PDEs towards the solution of the corresponding effective equation, as well as to describe small-scale oscillations and characterize fluctuations. From the mathematical perspective, it gives rise to a subtle mixture of probability and PDE analysis: the difficulty boils down to understanding how statistical properties of random coefficient fields are transmitted to the solution operator — a particularly nontrivial question as the solution operator is a nonlinear nonlocal transformation of the coefficients.
In this course, after an introduction to the qualitative theory of stochastic homogenization, we shall focus on the optimal quantitative theory that has been developed over the last decade in the model framework of linear elliptic equations with heterogeneous coefficients. We aim to give a broad presentation of the fundamental ideas of this new theory and to provide self-contained proofs of the optimal results in the simplest setting of weakly correlated Gaussian coefficients. We shall particularly emphasize the central role of large-scale regularity theory, which formalizes how improved regularity properties for heterogeneous PDEs can be inherited from the corresponding effective equation on large scales.
Prerequisite: functional analysis, probability theory, elliptic PDEs, elliptic regularity theory.
Contact: For any more precise information on this course, please write me at my e-mail address mduerinc(at)math.ucla.edu.
Table of contents:
I Introduction
1 Goals of the course
2 History
3 Main assumptions
4 Explicit 1D case
II Qualitative theory
1 Formal 2-scale expansion
2 Stationary differential calculus
3 Existence of correctors
4 Proofs of homogenization result
5 Qualitative corrector result
6 Towards a quantitative theory
III Quantitative theory of oscillations
1 Gaussian setting and Malliavin calculus
2 Annealed L^p regularity theory: perturbative version
3 Stochastic corrector estimates
4 Quantitative corrector result
IV Quantitative theory of fluctuations
1 Pathwise description of fluctuations
2 Scaling limit
V Large-scale regularity theory
1 Annealed Green's function estimates
2 Large-scale Schauder regularity
3 Large-scale and annealed L^p regularity
Schedule:
Course #1 (01/04, 12pm-1pm): Sections I.1-I.2 (see notes)
Course #2 (01/06, 12pm-1.15pm): Sections I.3-I.4 (see notes)
Course #3 (01/11, 12pm-1pm): Sections II.1-II.2 (see notes)
Course #4 (01/13, 12pm-1pm): Section II.2-II.3 (see notes)
Course #5 (01/20, 12pm-1pm): Section II.3-II.4 (see notes)
Course #6 (01/22, 12pm-1.15pm): Section II.4 (see notes)
Course #7 (01/25, 12pm-1pm): Section II.5-II.6 (see notes)
Course #8 (01/27, 12pm-1pm): Section III.1 (see notes)
Course #9 (01/29, 12pm-1pm): Section III.1-III.2 (see notes)
Course #10 (02/01, 12pm-1pm): Section III.1-III.2 (see notes)
Course #11 (02/03, 12pm-1pm): Section III.2 (see notes)
Course #12 (02/05, 12pm-1pm): Section III.2 (see notes)
Course #13 (02/08, 12pm-1pm): Section III.3 (see notes)
Course #14 (02/10, 12pm-1pm): Section III.3-III.4 (see notes)
Course #15 (02/12, 12pm-1pm): Section IV.1 (see notes)
Course #16 (02/17, 12pm-1.15pm): Section IV.1-IV.2 (see notes)
Course #17 (02/19, 12pm-1pm): Section IV.2 (see notes)
Course #18 (02/22, 12pm-1pm): Section V.1-V.2 (see notes)
Course #19 (02/24, 12pm-1pm): Section V.2 (see notes)
Course #20 (02/26, 12pm-1pm): Section V.2 (see notes)
Course #21 (03/03, 12pm-1pm): Section V.2 (see notes)
Course #22 (03/05, 12pm-1pm): Section V.3 (see notes)
Course #23 (03/08, 12pm-1pm): Section V.3 (see notes)