Quantitative unique continuation properties of elliptic PDEs with variable 2nd order coefficients and applications in control theory, Anderson localization, and photonics

本项目致力于研究大区域上偏微分方程解的不确定性关系及其应用。特别是，我们要证明唯一的连续估计和不确定性原则的范围内的频谱投影仪的椭圆微分算子的有界和无界矩形域上的可变二阶系数的功能。考虑到所需的应用，假设抽样集是等分布的，估计需要在矩形域类上是均匀的。这将允许我们处理数学物理和应用分析中的三个当今问题：（1）导出有界和无界区域上具有可变热扩散率的热传导的控制成本估计，（2）分析在光子晶体理论中至关重要的本征谱的带边的移动，最后（3）研究了模拟波在无序介质中传播的随机发散型算子，并建立了以前无法达到的无序区域中的安德森局部化

The project is devoted to the study of uncertainty relations for PDE solutions on large domains and applications thereof. In particular, we want to prove unique continuation estimates and uncertainty principles for functions in the range of spectral projectors of elliptic differential operators with variable second order coefficients on bounded and unbounded rectangular domains. In view of the desired applications, the estimates need to be uniform over the class of rectangular domains, provided that the sampling set is equidistributed.This will allow us to treat three present-day problems in mathematical physics & applied analysis:(1) derive control cost estimates for heat conduction with variable thermal diffusivity on bounded and unbounded domains,(2) analyse the movement of band edges of the essential spectrum, which is of crucial importance in the theory of photonic crystals, and finally(3) study random divergence type operators modelling propagation of waves in disordered media and establish Anderson localization in previously inaccessible disorder regimes