Spectral analysis of micro-resonant PDEs with random coefficients

When waves travel in the three-dimensional space in the absence of obstacles, their behaviour is fairly simple and very well understood. However, if one wants to propagate, say, electromagnetic or sound waves along a curved surface or through an inhomogeneous material, the problem becomes less straightforward and its mathematical description far trickier. The nontrivial geometry of the underlying space is reflected in both the physical properties of the propagating waves and the complexity of their mathematical modelling.

In the 1950s, Philip W. Anderson (Nobel Prize in Physics, 1977) realised that one can induce localisation of electrons (that is, electrons, which can be viewed as a particular kind of waves, live in a confined small portion of space, rather than propagate over extended regions) in a material with a lattice structure by adding a certain amount of randomness to the system, a phenomenon now known as Anderson localisation. This can be achieved, for example, by contaminating a semi-conductor with randomly distributed impurities. Despite the extensive mathematical and experimental efforts made since then to grasp the theoretical underpinning of wave localisation, this remains an elusive phenomenon and the mathematical techniques to describe it are few and far between.

The proposal deals with the rigorous mathematical description of propagation and localisation of waves in a particular class of composite materials with random microscopic geometry, called micro-resonant (or high-contrast) random media: small inclusions of a "soft" material are randomly dispersed in a "stiff" matrix. The highly contrasting physical properties of the two constituents, combined with a particular scaling of the inclusions, result in microscopic resonances, which manifest macroscopically by allowing propagation of waves in the material only within certain ranges of frequencies (band-gap spectrum) - a property quite useful in the manufacturing of wave manipulating devices.

High-contrast media with periodically distributed inclusions have been extensively studied and numerous results are available in the literature. However, their stochastic counterparts, which model more realistic scenarios and may exhibit localisation, are very little understood from a mathematical viewpoint. The proposal will develop a new range of techniques to study Anderson-type localisation and defect modes in the context of composite materials modelled by high-contrast partial differential equations with random coefficients. The proposed new approach, based on the interplay between spectral theory and stochastic homogenisation, is exciting and very promising, in that it links the mathematical techniques with the underlying localisation mechanism due to the micro-resonant effect of inclusions. The project will also develop a comprehensive homogenisation and spectral theory for high-contrast random systems of PDEs (describing, for example, electromagnetic and elastic waves), for which nothing is currently known, and which have the potential of giving rise to new previously unobserved effects.