Elliptic systems in irregular domains: asymptotics, spectral theory and homogenisation

我们周围的大多数现象都是由数学方程描述的，特别是所谓的偏微分方程（PDE），甚至同时由几个偏微分方程（即PDE系统）描述。这种现象的一个例子是由线性弹性方程组描述的弹性体的变形或振动。（另一个例子是由麦克斯韦电磁方程描述的无线电波的传播。）数学上理解这类系统的解的行为对于描述和预测各种物理效应是很重要的。解的数学性质又取决于系统的类型（在我们的例子中，系统是椭圆型的），但重要的是，取决于边界的结构和所谓的边界条件的性质，边界上的附加数学方程来自物理。一些边界是规则的（即漂亮和光滑），但其他边界可能是不规则的，包含例如角落，裂缝，尖峰等。不规则边界的存在一方面会引起有趣的数学问题，但另一方面可能会导致新的物理现象，例如薄零件的显著弯曲及其非常缓慢的振动。在这个项目中，我们的目标是发现和分析的数学性质的解决方案，这样的系统中存在的强不规则性。我们观察到，这样的系统的某些基本数学性质失败，如紧凑性。这使得某些经典的数学技术的应用是不可能的，并提出了开发替代的，更先进的技术的挑战。我们专注于开发这些技术，这些技术本身是一个有趣的数学，但也有好奇的进一步的影响和应用。因此，这个应用驱动的项目希望是一个融合了深刻和严格的数学与其物理和机械应用。

Most of the phenomena around us are described by mathematical equations, in particular by the so-called partial differential equations (PDEs) or even by several of those simultaneously (i.e. by a system of PDEs). An example of such a phenomenon is deformation or vibration of an elastic body, described by a system of equations of linear elasticity. (Another example is propagation of radiowaves described by Maxwell's equations of electromagnetism.) Mathematical understanding of the behaviour of solutions of such systems is important for describing and predicting various physical effects. Mathematical properties of the solutions depend in turn on the type of a system (in our examples the systems are of elliptic type), but also, importantly, on the structure of the boundary and on the nature of the so-called boundary conditions , additional mathematical equations on the boundary from physics. Some boundaries are regular (i.e. nice and smooth) but others may be irregular , containing for example corners, cracks, spikes , etc. The presence of irregular boundaries on one hand poses interesting mathematical problems but on the other hand may cause new physical phenomena like for example a significant bending of thin parts and their very slow vibrations. In this project we aim at discovering and analysing mathematical properties of solutions of such systems in the presence of strong irregularitiers. We observe that then certain basic mathematical properties of such systems fail, such as compactness. This makes application of certain classical mathematical techniques impossible, and poses the challenge of developing alternative, more advanced, techniques. We concentrate on developing those techniques, which themselves are a piece of interesting mathematics but also have curious further implications and applications. Hence, this applications driven project is hoped to be a blend of deep and rigorous mathematics with its physical and mechanical applications.