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

该项目围绕着一种特殊类型的所谓“复合”材料的研究。一般来说，复合材料是通过以某种方式“混合”两种或多种基本材料来获得的，以获得上级混合物中单个组分的性能。复合材料的例子包括碳纤维树脂、钢筋混凝土、光纤电缆和桁架桥。虽然许多现有的复合材料是通过试验和错误获得的，但数学已被证明是理论上预测其性能的强大且具有成本效益的工具。通常，复合材料被认为是一种“连续”介质，并通过偏微分方程（PDE）进行建模，其解代表一些感兴趣的物理量。因此，这种偏微分方程的数学分析是理解复合材料的行为的关键。本项目研究一类偏微分方程，迄今为止已经达到的数学理论。分析这些方程的潜在困难表现在组分性质之间的高度对比以及其中一个组分（复合物的“子”）占据非常小的体积的事实。这两个特征需要开发新的分析工具，这些工具从数学的角度来看是有价值的，同时导致了许多关于复合材料的明确结论，这些结论可以被物理学家，材料科学家，电气和机械工程师利用。该项目的成果将立即用于具有实际意义的领域，如先进通信设备的设计和最先进的结构工程。它们将涉及对上述复合材料种类的定性描述以及在实际实施中可以依赖的定量估计。

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.