**Pattern formation, energy landscapes, and scaling laws**

**Head: **

Felix Otto

**Contact: **Email

*Phone:*

+49 (0) 341 - 9959 - 950

**Address:**

Inselstr. 22

04103 Leipzig

**Administrative Assistant:**

Katja Heid

Email, *Phone/Fax:*

+49 (0) 341 - 9959

- 951

- 658

# Lecture in the Winter term 2014 / 2015

## Stochastic homogenization and elliptic regularity theory

**Lecturer:**Felix Otto**Date:**Tuesday 09.15 - 11.00**Room:**MPI MiS, A 01**Language:**English**Target audience:**MSc students, PhD students, Postdocs**Content (Keywords):**elliptic partial differential equations, random regularity theory**Prerequisites:**This course requires less technology in probability theory than it seems and will be self-contained in that respect. On the other hand, the course gives a good opportunity to recapitulate some classical techniques of elliptic regularity theory, like the approach to Schauder theory via Campanato spaces.

## Abstract

In many applications, one has to solve a linear elliptic partial differential equation with uniformly elliptic coefficients that vary on a length scale much smaller than the domain size. We are interested in a situation where the coefficients are characterized in *stochastic* terms: Their statistics are assumed to be translation invariant and to decorrelate over large distances. As is known since more than forty years, the solution operator behaves - on large scales - like the solution operator of an elliptic problem with *homogeneous* deterministic coefficients!

A more recent insight is that, on large scales and with high probability, the regularity properties of solutions are very close to those of an equation with homogeneous coefficients, for instance in terms of Liouville-type statements. I will focus on this "random regularity theory", which turns out to be much stronger than the deterministic one in the class of uniformly elliptic coefficients, especially in case of systems.