Mathematical foundations of metamaterials: homogenisation, dissipation and operator theory

为了了解各种物理介质在特定条件下的行为，例如：a)地震期间地球表面如何变形，或b)使用光纤内窥镜可以达到多高的图像分辨率，数学家编写了微分方程式(DES)并研究其解的分析性质，然后对其进行解释，以得出关于现实生活对象的结论。此活动的一部分涉及对另外依赖于参数的DES的分析。在上述实例中，该参数可以是：a)成岩晶体的大小与地面中岩层厚度的比率；b)内窥镜的厚度与长度的比率。该项目将根据“算符”及其谱的数学理论的最新成果，开发一种分析依赖于参数的DES解的新方法，这些谱大致可被认为是算符“值”的集合。一般说来，算符的谱是在“复”数的二维平面上找到的。然而，对于许多表示DES的运算符来说，谱是该平面中直线的子集。事实证明，将这样一个算子视为更广泛的算符家族的成员，其频谱不一定位于同一条直线上，会带来许多技术上的好处，类似地，在复平面上的分析有助于理解实数。在过去的50年左右的时间里，通过使用这种类比，已经得到了许多关于算子(以及作为其特例的DES)的漂亮的数学结果。我们将利用这些结果来提高我们对DES家庭行为的理解。作为这类族的一个特殊来源，我们将研究表示复合材料的方程，即具有几个较简单组成部分的介质。我们周围的许多物体都是复合材料，例如，木材、多孔岩石、泡沫、起泡液体、增强树脂、多晶金属。我们所针对的数学陈述将提供有关这些现实生活中的物体的新信息，例如，它们的声学特性，或者它们与电磁场相互作用的方式。从物理学的角度来看，这个更广泛的运算符家族的成员承认，与最初的“无损失”设置相比，存在一些耗散(即能量损失)。该项目将在DES描述复合材料的情况下为这种耗散扩展提供一个通用的数学框架，从而产生一种新的分析方法来研究它们的有效性质。

In order to understand how various physical media behave under specific conditions, for example: a) how the earth surface is deformed during an earthquake, or b) what image resolution one can achieve with a fibre-optic endoscope, mathematicians write differential equations (DEs) and study analytical properties of their solutions, which are then interpreted to draw conclusions about real-life objects. Part of this activity involves analysis of DEs that additionally depend on a parameter. In the above examples this parameter could be: a) the ratio of the size of the rock-forming crystals to the thickness of layers of rock in the ground, orb) the thickness-to-length ratio of the endoscope. The project will develop a new approach to the analysis of solutions of parameter-dependent DEs, based on recent achievements in the mathematical theory of ``operators'' and their spectra, which could roughly be thought of as the sets of the operator ``values''. In general, the spectrum of an operator is found in the two-dimensional-plane of ``complex'' numbers. However, for many operators representing DEs, the spectrum is a subset of a straight line in this plane. It turns out that considering such an operator as a member of a wider operator family, whose spectra are not necessarily situated on the same line, brings about a lot of technical benefits, in a similar way analysis in the complex plane helps understanding real numbers. In the last 50 years or so, many elegant mathematical results about operators (and about DEs as their particular case) have been obtained by using this analogy. We will exploit these results in order to improve our understanding of the behaviour of families of DEs. As a particular source of such families we will study equations representing composites, i.e. media that have several simpler constituent parts. Many objects around us are composites, for example, wood, porous rocks, foams, bubbly liquids, reinforced resins, polycrystal metals. Mathematical statements that we aim at will provide new information about such real-life objects concerning, for example, their acoustic properties, or the way in which they interact with an electromagnetic field. From the physics point of view, members of this wider operator family admit some dissipation (i.e. loss of energy) in comparison to the original ``loss-free'' setup. The project will provide a general mathematical framework for such dissipative extensions in the case of DEs describing composites, yielding a new analytic approach to the study of their effective properties.