  This paper presents and compares three analytical methods for calculating low
frequency band gap boundaries in doubly periodic arrays of resonating thin
elastic shells. It is shown that both lattice sum expansions in the vicinity of
its poles and self consistent scheme could be used to predict boundaries of
low-frequency band gaps due to axysimmetric (n=0) and dipolar (n=1) shell
resonances. The accuracy of the former method is limited to low filling
fraction arrays, however the application of the matched asymptotic expansions
could significantly improve approximations of the upper band gap boundary due
to axysimmetric resonance. The self-consistent scheme is shown to be very
robust and gives reliable results even for dense arrays with filling fraction
higher than 40%. The results are used to predict the dependence of the position
and width of the low frequency band gap on the properties of shells and their
periodic arrays.
