Spectral analysis of micro-resonant PDEs with random coefficients

当波在没有障碍物的三维空间中传播时，它们的行为相当简单，而且非常容易理解。然而，如果一个人想要沿着曲面或通过非均匀材料传播电磁波或声波，这个问题就变得不那么简单了，它的数学描述也变得复杂得多。在20世纪50年代，菲利普·W·安德森(1977年诺贝尔物理学奖)认识到，人们可以通过在具有晶格结构的材料中添加一定的随机性来诱导电子的局域化(即电子，可以被视为一种特殊的波，而不是在扩展的区域中传播)。例如，这可以通过用随机分布的杂质污染半导体来实现。尽管从那时起，人们进行了大量的数学和实验工作来掌握波局部化的理论基础，但这仍然是一个难以捉摸的现象，描述它的数学技术也很少。该提议涉及到波在具有随机微观几何形状的特定类型的复合材料中传播和局部化的严格数学描述，称为微共振(或高对比度)随机介质：微小的软材料夹杂物随机分散在一个刚性的矩阵中。这两种成分的高度对比的物理性质，结合夹杂物的特定比例，导致微观共振，这在宏观上表现为仅允许波在特定频率范围内(带隙频谱)在材料中传播-这一特性在波操纵装置的制造中非常有用。具有周期性分布夹杂物的高对比度介质已被广泛研究，并在文献中获得了大量结果。然而，它们的随机对应物模拟了更现实的场景，并可能表现出局部化，从数学角度来看，人们很少理解它们。这项提议将开发一系列新的技术，在复合材料的背景下研究安德森类型的局域模和缺陷模，该复合材料由具有随机系数的高对比度偏微分方程组建模。提出的新方法基于谱理论和随机均一化之间的相互作用，是令人兴奋和非常有前途的，因为它将数学技术与由于夹杂物的微共振效应而产生的潜在局域化机制联系起来。该项目还将为高对比度的PDE随机系统(例如，描述电磁波和弹性波)开发全面的均化和频谱理论，目前对这些系统一无所知，并有可能产生新的以前未观察到的影响。

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.