Stochastic Homogenisation of finite Partitions: QualitativeTheory and Quantitative Aspects

Annika Bach (Technische Universität München) #

This talk is concerned with the homogenisation of random integral functionals defined on finite partitions, that is, functions of bounded total variation and taking only finitely many values. In the first part of the talk we will recall the notion of stationary and ergodic integrands together with some techniques which can be applied to prove a qualitative homogenisation result in the framework of Γ-convergence. We will see in particular that the Γ-limit becomes deterministic and the limiting integrand is obtained via an asymptotic cell formula by solving suitable minmisation problems on larger and larger cubes. In the second part of the talk we will show how to control quantitatively (in terms of the cube size) the fluctuations of these cell formulas. As a byproduct we obtain a simplified cell formula where the cubes are replaced by almost flat hyperrectangles. This is joint work with Matthias Ruf (EPFL).