Mathematical foundations of metamaterials: homogenisation, dissipation and operator theory

点击翻译按钮获取中文摘要

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.