The mathematical analysis and applications of a new class of high-contrast phononic band-gap composite media

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The project revolves around the study of so-called ``composite'' materials of a special kind. In general, composites are obtained by ``mixing'' in some way two or more basic materials, in order to achieve properties that are superior to those of the individual components of the mixture. Examples of composites include carbon fibre resins, reinforced concrete, optical fibre cables and truss bridges. While many existing composites have been obtained experimentally by trial and error, mathematics has proved to be a powerful and cost-efficient tool in predicting their properties theoretically. Usually a composite is thought of as a ``continuous'' medium and is modelled by means of a partial differential equation (PDE), whose solution represents some physical quantity of interest. The mathematical analysis of such PDEs is thus key in understanding the behaviour of composites The present project investigates a class of PDEs that have thus far been out of reach of the mathematical theory. The underlying difficulties in the analysis of these equations are presented by a high degree of contrast between the properties of the constituents and the fact that one of them (the ``skeleton'' of the composite) occupies a very small volume. These two features require the development of new analytical tools that are valuable from the mathematical perspective and at the same time lead to a host of explicit conclusions about composites, which can be utilised by physicists, material scientists, electrical and mechanical engineers. The results of the project will be of immediate use in areas of practical interest, such as the design of advanced communication devices and state-of-the-art structural engineering. They will involve a qualitative description of the kind of composites described above and quantitative estimates that can be relied upon in their practical implementation.