Spectral analysis of micro-resonant PDEs with random coefficients

具有随机系数的微共振偏微分方程的谱分析

• 批准号: EP/X01021X/1
• 负责人: Matteo Capoferri
• 金额: $ 40.45万
• 依托单位: Heriot-Watt University
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX01021X%2F1
• 关键词: Spectral; analysis; micro; resonant; PDEs

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.

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，也很容易理解。然而，如果人们想传播，比如说，电磁波或声波沿着一个弯曲的表面或通过一个不均匀的材料，这个问题就变得不那么简单了，它的数学描述也要复杂得多。基础空间的非平凡几何结构反映在传播波的物理性质和数学建模的复杂性上。安德森（1977年诺贝尔物理学奖）意识到，人们可以通过向具有晶格结构的材料中添加一定量的随机性来诱导电子的局域化（也就是说，电子，可以被视为一种特殊类型的波，生活在有限的小部分空间中，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，人们在数学和实验上进行了大量的努力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。该提案涉及波在具有随机微观几何形状的特定类别复合材料中的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：“软”材料的小夹杂物随机分散在“硬”基质中。这两种成分的物理性质差异很大，再加上夹杂物的特定比例，导致了微观共振，其通过仅允许在特定频率范围内的波在材料中传播而在宏观上显现（带隙谱）-在波操纵装置的制造中非常有用的性质。高-具有周期性分布的内含物的造影剂已经被广泛研究，并且在文献中可以获得许多结果。然而，它们的随机对应物，模拟更现实的场景，并可能表现出局部化，从数学的角度来看，很少被理解。该提案将开发一系列新的技术，以研究安德森型局部化和缺陷模式的背景下，复合材料建模的高对比度偏微分方程与随机系数。所提出的新方法，光谱理论和随机均匀化之间的相互作用的基础上，是令人兴奋的和非常有前途的，因为它链接的数学技术与底层的本地化机制，由于微观共振效应的夹杂物。该项目还将为偏微分方程的高对比度随机系统（例如，描述电磁波和弹性波）开发全面的均匀化和谱理论，目前对此一无所知，并且有可能产生新的以前未观察到的效应。