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

本课题致力于研究大域PDE解的不确定性关系及其应用。特别地，我们想证明在有界和无界矩形区域上变二阶系数椭圆微分算子的谱投影范围内函数的唯一延拓估计和不确定性原理。考虑到期望的应用，假设采样集是等分布的，估计需要在矩形域上是均匀的。这将使我们能够处理当前数学物理和应用分析中的三个问题：(1)在有界和无界域上导出具有可变热扩散率的热传导的控制成本估算；(2)分析在光子晶体理论中至关重要的基本光谱的带边运动；最后(3)研究随机散度型算子模拟波在无序介质中的传播，并在先前不可达的无序状态中建立安德森局部化

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