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

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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