An example of multiple gaps in the spectrum of a periodic waveguide

A periodic waveguide is constructed, whose shape depends on a small parameter $h>0$, in which the essential spectrum of the operators for some boundary-value problems (Dirichlet, Neumann, and mixed under certain restrictions) for a formally self-adjoint elliptic system of second-order differential equations acquires any pre-assigned number of gaps. The geometric shape of the waveguide can be interpreted as an infinite periodic set of beads connected by thin, short ligaments. The proof of that gaps appear is based on an application of the max-min principle and the weighted Korn inequality.
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