Quantitative elliptic homogenization: at the interface of PDEs, probability theory and statistical physics

20.11.2024, 14:00 - 16:00 (Special session)  –  Campus Golm, Building 9, Room 2.22 and via Zoom
Institutskolloquium

Mitia Duerinckx (Brussel), Marius Lemm (Tübingen)

14:00  Mitia Duerinckx (Bruxelles):  Quantitative elliptic homogenization: a regularity perspective and Bourgain's surprising insight.
14:45  Tea and Coffee Break
15:15  Marius Lemm (Tübingen):  Quantitative elliptic homogenization: an harmonic-analysis approach.

Mitia Duerinckx (Université Libre de Bruxelles):  Quantitative elliptic homogenization: a regularity perspective and Bourgain's surprising insight.

Abstract:     Homogenization theory has been developed as a rigorous background for the study of macroscopic effective properties of heterogeneous media in materials sciences. In the first talk, we start by reviewing the state-of-the-art of the theory: we describe the expected fine description of heterogeneous solution operators by means of so-called two-scale expansions and we emphasize the fundamental limitations of such descriptions in case of random (vs. periodic) heterogeneities. We then turn to an alternative formulation of homogenization problems as regularity questions for some pseudo-differential symbols. This reformulation remained quite confidential until it was used by Bourgain in 2017 to obtain perturbatively a surprisingly strong improvement of some of the available results in the field. We will emphasize how surprising this result was at first and how incompatible it is with standard two-scale approaches. Some applications will be discussed, as well as a parallel use of those ideas for problems in statistical physics.
    

Marius Lemm (Universität Tübingen):  Quantitative elliptic homogenization: an harmonic-analysis approach

Abstract:     In the second talk, we further describe Bourgain's approach to homogenization questions, which has recently led to surprisingly improved results in a perturbative setting. The initial idea, due to Sigal, is to represent the averaged Green's function via the Schur complement formula as an infinite perturbation series. To control the perturbation series, one needs to simultaneously exploit random and oscillatory (harmonic-analytic) cancellations. ***The problem raised by a lack of compatibility between these two types of cancellation was solved in 2017, by Bourgain who found a solution in a discrete i.i.d. setting by means of a new disjointification trick.*** It was recently shown that this approach is highly robust, i.e., it extends to continuum systems, including those with long-range correlations. Moreover, it turns out that for Gaussian-type random variables, Malliavin calculus can replace the disjointification trick. Based on joint works with Mitia Duerinckx, Antoine Gloria, Jongchon Kim, and François Pagano.
   

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