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

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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.