Homogenization of elliptic PDE in divergence-form: large-scale regularity and quantitative results

Homogenization concerns the large-scale behavior of the solutions of PDE with heterogeneous coefficients. This course will treat the model problem consisting of linear PDE in divergence form with stationary and ergodic (e.g. periodic, quasi-periodic, or random) coefficients. It consists of three parts. The first part will be a gentle introduction to the subject in the context of periodic coefficients, presenting the variational proof by G-convergence, explaining the role of correctors and the two-scale expansion, and briefly discussing boundary layer effects. The second part will present the qualitative large-scale regularity theory of Gloria, Neukamm, and Otto, which shows that averaging effects lead to better-than-expected regularity at large scales. The final part treats quantitative error estimates following the recent book of Armstrong and Kuusi: the main thrust of the work will lie in quantifying the variational argument from the first part of the course through a combination of PDE arguments and concentration inequalities.

Date and time info
Tuesday 14.00-15.30

Keywords
partial differential equations, homogenization, regularity theory

Prerequisites
basic PDE (e.g. Evans, up to and including Chapter 6), knowledge of probability is helpful later