Kirill Cherednichenko (Cardiff)

Homogenisation of elliptic PDE and the associated techniques

Abstract:
I will give a brief introduction to the analysis of elliptic PDE with rapidly oscillating coefficients, discussing some elements of this theory developed in the 1970's such as the energy method by L. Tartar. I will then move on to the more recent techniques for homogenisation of non-uniformly elliptic problems, in particular those where the coefficients have a high degree of contrast between the values in different parts of the periodicity cell. From the point of view of spectral analysis, one can show the Hausdorff convergence of the spectra of the related family of PDO to the spectrum of a certain two-scale ``homogenised'' operator, which has a band-gap structure (result by V. Zhikov). If time permits I will also mention some homogenisation methods in the context of calculus of variations.