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，很容易理解。然而，如果一个人想要传播，比如说，电磁波或声波沿着曲面或穿过非均匀材料，问题就变得不那么简单了，它的数学描述也要复杂得多。底层空间的非平凡几何既反映在传播波的物理性质上，也反映在其数学模型的复杂性上。20世纪50年代，菲利普·w·安德森（Philip W. Anderson， 1977年诺贝尔物理学奖得主）意识到，在晶格结构的材料中，通过向系统中添加一定数量的随机性，可以诱导电子的局域化（也就是说，电子可以被视为一种特殊的波，生活在有限的一小部分空间中，而不是在扩展的区域中传播），这种现象现在被称为安德森局域化。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，人们进行了广泛的数学和实验努力，以掌握波局域化的理论基础，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。该提案处理了波在一类具有随机微观几何结构的特殊复合材料中的传播和定位的严格数学描述，称为微共振（或高对比度）随机介质：“软”材料的小内含物随机分散在“硬”矩阵中。两种成分的高度对比的物理特性，加上夹杂物的特定缩放，导致微观共振，宏观上表现为只允许波在材料中的某些频率范围内传播（带隙频谱）-这一特性在制造波操纵设备中非常有用。具有周期性分布包裹体的高对比度介质已被广泛研究，并在文献中获得了许多结果。然而，从数学的角度来看，它们的随机对应物（模拟更现实的场景并可能表现出局部性）很少被理解。该提案将开发一系列新的技术来研究anderson型局部化和缺陷模式，这些模式是由具有随机系数的高对比度偏微分方程建模的复合材料。提出的新方法，基于谱理论和随机均质化之间的相互作用，是令人兴奋和非常有前途的，因为它将数学技术与由于包含物的微共振效应而导致的潜在定位机制联系起来。该项目还将为pde的高对比度随机系统（例如，描述电磁波和弹性波）开发一个全面的均匀化和光谱理论，目前对这些系统一无所知，并且有可能产生以前未观察到的新效应。

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.