Abstract:

The linear elasticity problem is an elliptic system of PDE's in a given domain, associated with Neumann-type boundary conditions. We consider some domains with nontrivial geometries, which lead to interesting spectral structure of the problem. One example concerns a bounded body with cuspidal, beak-like shape. We characterize the discreteness of the spectrum in terms of the sharpness of the beak. In particular, given a sharp enough shape, the essential (and continuous) spectra are not empty. Another case deals with an unbounded periodic waveguide. We construct examples which have band-gap like essential spectrum and sketch a method using which one can prove the existence of any given number of spectral gaps. This reports on joint work with Sergey Nazarov and others.