Homogenisation of Multiscale Partial Differential Equations from Physics

Research areas: Continuum mechanics, Mathematical analysis, Mathematical physics and Material Engineering - composites

The project will focus on advancing a general theory of ''non-classical'' homogenisation for partial differential equations (PDEs) coming from Physics, typically with a ''strong'' interaction between the scales. One general scenario relates to strongly heterogeneous media, which may in particular be electromagnetic, acoustic or elastic, where the contrast between the constituent components is high. As a result, some component may respond to an applied dynamic frequency as resonators while others will essentially behave as in a quasi-static regime amenable to a classical homogenisation. Adopting appropriate scaling results in a high-contrast homogenisation problem for relevant PDE, where the two small parameters of the contrast and of the microscale of the heterogeneity are critically scaled. The nature of such a coupling between the scales leads to a number of interesting physical effects at the macroscale. These include frequency bandgaps, high dispersion, wave localisation, etc. The project will be aiming at advancing mathematical understanding of these effects, which would require development of non-classical tools of multi-scale asymptotic analysis of PDEs. Precise topics are envisaged to evolve, but some specific topics may include: analysing the effect of ''coupled resonances'' in high-contrast periodic media with a pre-resonant propagating frequency; investigating the effect of randomness of high-contrast resonators on wave localisation due to trapping by the resonators with statistically distributed eigenfrequencies; analysing the effects of directional vs frequency filtering properties for more general patterns of asymptotic degeneracies including high anisotropies and particularly for vector problems such as in electromagnetism and elastodynamics; high-contrast lattice materials; homogenisation of small-size resonators via capacity-type tools. One additional intriguing challenge is to attempt to develop a homogenisation theory approach to Einstein's equations of General Relativity.