Jari Taskinen (Helsinki)
Remarks on the spectrum of the linearized elasticity problem
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.